Physics Book - User contributions [en]

Point Charge

2020-12-10T20:34:38Z

Amummareddy3: /* The Main Idea (CLAIMED BY ASHITA MUMMAREDDY FALL 2020) */

This page is all about the [[Electric Field]] due to a Point Charge.

==The Main Idea ==
(Ch 13.1 in ''Matter & Interactions Vol. 2: Modern Mechanics, 4th Edition by R. Chabay & B. Sherwood'')

'''Point Charge/Particle''' - an object with a radius that is very small compared to the distance between it and any other objects of interest in the system. Since it is very small, the object can be treated as if all of its charge and mass are concentrated at a single "point".
*Electrons and Protons are always considered to be point particles unless stated otherwise

<u> 2 types of point charges: </u>
*Protons (e) --> positive point charges, ( q = 1.6e-19 Coulombs)
*Electrons (-e) --> negative point charges, (q = -1.6e-19 Coulombs)

''Like'' point charges ''attract'', ''opposite'' point charges ''repel''.
ex.<table border> <tr>
<th> Point Charges </th>
<th> Result </th>
<th>Diagram</th>
</tr>
<tr>
<td> 1 proton, 1 electron</td>
<td> Attract </td>
<td>[[File:Proton_electron_attraction.png]]</td>
</tr>
<tr>
<td>2 protons </td>
<td> Repel </td>
<td>[[File:Proton_repulsion.png]]</td>
</tr>
<tr>
<td>2 electrons </td>
<td> Repel </td>
<td>[[File:Electron_repulsion.png]]</td>
</tr>
</table border>

===The Electric Field===
(Ch 13.3 in ''Matter & Interactions Vol. 2: Modern Mechanics, 4th Edition by R. Chabay & B. Sherwood'')

The electric field created by a charge is present throughout space at all times, whether or not there is another charge around to feel its effects.

Electric Field of a Charge Observed at a location: F = Eq
*F = Force on particle 2
*E = electric field at source location
*q = charge of particle 2
The magnitude of the electric field decreases with increasing distance from the point charge.

<table border>
<tr>
<td>The electric field of a positive point charge points radially outward</td>
<td>The electric field of a negative point charge points radially inward</td>
</tr>
<tr>
<td>[[File:Proton_electric_field.png]] </td>
<td> [[File:Electron_electric_field.png]] </td>
</tr>
</table border>

===A Mathematical Model===

====Electric Field due to Point Charge====
(Ch 13.4 in ''Matter & Interactions Vol. 2: Modern Mechanics, 4th Edition by R. Chabay & B. Sherwood'')

Electric Field of a Point Charge (<math>\vec E</math>):

<math>\vec E=\frac{1}{4 \pi \epsilon_0 } \frac{q}{\mid\vec r\mid ^2} \hat r</math> (Newtons/Coulomb)

*<math>\frac{1}{4 \pi \epsilon_0 } </math> is Coulomb's Constant and is approximately <math>8.987*10^{9}\frac{N m^2}{C^2} </math>
*'''''q''''' is the charge of the particle
*'''''r''''' is the magnitude of the distance between the observation location and the source location
*<math>\hat r </math> is the unit vector in the direction of the distance from the source location to the observation point.

The direction of the electric field at the observation location depends on the both the direction of <math>\hat r </math> and the sign of the source charge.
*If the source charge is positive, the field points away from the source charge.
*If the source charge is negative, the field points toward the source charge.

====Coulomb Force Law for Point Charges====
(Ch 13.2 in ''Matter & Interactions Vol. 2: Modern Mechanics, 4th Edition by R. Chabay & B. Sherwood'')

<math>\mid\vec F\mid=\frac{1}{4 \pi \epsilon_0 } \frac{\mid Q_1Q_2 \mid}{r^2}</math>

Coulomb's law is one of the four fundamental physical interactions, and it describes the magnitude of the electric force between two point-charges.
<math>Q_1, Q_2</math>= The charge of two particles of interest

*<math>\mid\vec F\mid=\frac{1}{4 \pi \epsilon_0 }</math> = constant,
* <math>Q_1, Q_2</math> = the magnitudes of the point charges
*r = The distance between the two particles

====Connection Between Electric Field and Force====
The force on a source charge is determined by <math> F = Eq </math> where '''''E''''' is the electric field and '''''q''''' is the charge of a test charge in Coulombs.

By solving for the electric field in <math> F = Eq </math>, with F modeled by Coulomb's Law, you get the equation for the electric field of the point charge:

<math> E = \frac{F}{q_2} = \frac{1}{4 \pi \epsilon_0 } \frac{q_1q_2}{r^2}\frac{1}{q_2}\hat r = \frac{1}{4 \pi \epsilon_0 } \frac{q_1}{r^2} \hat r </math>

===A Computational Model===

Below is a link to a code which can help visualize the Electric Field at various observation locations due to a proton. Notice how the arrows decrease in size by a factor of <math> \frac{1}{r^{2}} </math> as the observation location gets farther from the proton. The magnitude of the electric field decreases as the distance to the observation location increases.

[[File:First code.gif]]

Two adjacent point charges of opposite sign exhibit an electric field pattern that is characteristic of a dipole. This interaction is displayed in the code below. Notice how the electric field points towards the negatively charged point charge (blue) and away from the positively charged point charge (red).

[[File:Code_2.png]]

==Examples==

===Simple===
There is an electron at the origin. Calculate the electric field at <4, -3, 1> m.

<table border>
<tr>
<td> <b>Step 1.</b> Find <math>\hat r</math>
Find <math>\vec r_{obs} - \vec r_{electron}:

((4,-3,1) - (0,0,0) = <4,-3,1> </math>m.

Calculate the magnitude of <math>\vec r</math>:

(<math>\sqrt{4^2+(-3)^2+1^2}=\sqrt{26}</math> From <math>\vec r</math>, find the unit vector <math>\hat{r}.</math> <math> <\frac{4}{\sqrt{26}},\frac{-3}{\sqrt{26}},\frac{1}{\sqrt{26}}> </math>
</td></tr>

<tr><td>'''Step 2:''' Find the magnitude of the Electric Field

<math> E_{mag}= \frac{1}{4 \pi \epsilon_0 } \frac{q}{r^2} = \frac{1}{4 \pi \epsilon_0 } \frac{-1.6 * 10^{-19}}{26} </math> </td>
<td> </td></tr>

<td>'''Step 3:''' Multiply the magnitude of the Electric Field by <math>\hat{r}</math> to find the Electric Field
<math>E = \frac{1}{4 \pi \epsilon_0 } \frac{-1.6 * 10^{-19}}{26}*<\frac{4}{\sqrt{26}},\frac{-3}{\sqrt{26}},\frac{1}{\sqrt{26}}>

= <-4.34*10^{-11},3.26*10^{-11},-1.09*10^{-11}> N/C </math></td>
</table border>

===Middling===
A particle of unknown charge is located at <-0.21, 0.02, 0.11> m. Its electric field at point <-0.02, 0.31, 0.28> m is <math><0.124, 0.188, 0.109> </math> N/C. Find the magnitude and sign of the particle's charge.

Given both an observation location and a source location, one can find both r and <math>\hat{r}</math> Given the value of the electric field, one can also find the magnitude of the electric field. Then, using the equation for the magnitude of electric field of a point charge,<math> E_{mag}= \frac{1}{4 \pi \epsilon_0 } \frac{q}{r^2} </math> one can find the magnitude and sign of the charge.

<table border>
<tr>
<td> <b>Step 1.</b> Find <math>\vec r_{obs} - \vec r_{particle} </math>:

<math>\vec r = <-0.02, 0.31, 0.28> m - <-0.21, 0.02, 0.11> m = <0.19,0.29,0.17> m </math>

To find <math>\vec r_{mag} </math>, find the magnitude of <math><0.19,0.29,0.17></math>

<math>\sqrt{0.19^2+0.29^2+0.17^2}=\sqrt{0.1491}= 0.39</math>
</td></tr>

<tr><td><b>Step 2:</b> Find the magnitude of the Electric Field:

<math>E= <0.124, 0.188, 0.109> N/C</math>

<math>E_{mag} = (\sqrt{0.124^2+0.188^2+0.109^2}=\sqrt{0.0626}=0.25</math> </td></tr>

<td>'''Step 3:''' Find '''''q''''' by rearranging the equation for <math>E_{mag}</math>

<math> E_{mag}= \frac{1}{4 \pi \epsilon_0 } \frac{q}{r^2} </math>

By rearranging this equation we get

<math> q= {4 pi * ε_0 } *{r^2}*E_{mag} </math>

<math> q= {1/(9*10^9)} *{0.39^2}*0.25 </math>

<math> q= + 4.3*10^{-12} C </math></td>

</table border>

===Difficult===
The electric force on a -2mC particle at a location (3.98 , 3.98 , 3.98) m due to a particle at the origin is <math>< -5.5*10^{3} , -5.5*10^{3}, -5.5*10^{3}></math> N. What is the charge on the particle at the origin?

Given the force and charge on the particle, one can calculate the surrounding electric field. With this variable found, this problem becomes much like the last one.
<math> E_{mag}= \frac{1}{4 \pi \epsilon_0 } \frac{q}{r_{mag}^2} </math> to find the rmag value. To find <math>\hat r</math> we can find the direction of the electric field as that is obviously going to be in the same direction as <math>\hat r</math>. Then, once we find <math>\hat r</math>, all that is left to do is multiply <math>\hat r</math> by rmag and that will give us the <math> r</math> vector. We can then find the location of the particle as we know <math>r=r_{observation}-r_{particle}</math>

<table border>
<tr>
<td> <b>Step 1.</b> Find the magnitude of the Electric field:
<math> F = Eq </math>
<math> <5.5e3, -7.6e3, 0> = E * -2mC </math>
<math> E = \frac{< -5.5e3 , -5.5e3, -5.5e3>}{-2mC}

= <2.75e6 , 2.75e6, 2.75e6> </math> N/C
</td></tr>

<tr><td><b>Step 2:</b> Find <math>\vec r_{obs} - \vec r_{particle} </math>.

<math>\vec r = <3.98 , 3.98 , 3.98> m - <0 , 0 , 0> m = <3.98 , 3.98 , 3.98> m </math>

To find <math>\vec r_{mag} </math>, find the magnitude of <math><3.98 , 3.98 , 3.98></math>

<math>\sqrt{3.98^2+3.98^2+3.98^2}=\sqrt{47.52}= 6.9</math>
</td></tr>

<td> '''Step 4:''' Find '''''q''''' by rearranging the equation for <math>E_{mag}</math>

<math> E_{mag}= \frac{1}{4 \pi \epsilon_0 } \frac{q}{r^2} </math>

By rearranging this equation we get

<math> q= {4 pi * ε_0 } *{r^2}*E_{mag} </math>

<math> q= {1/(9e9)} *{6.9^{2}}*4.76e6} </math>

<math> q= + 0.253 C </math></td>

</table border>

==Connectedness==
''1. How is this topic connected to something that you are interested in?''

I think the topic is interesting because electric fields are inside the human body and we self-create these fields constantly. It's really cool to think about how your body might be responding and creating these fields continuously and what happens when the balances are thrown off by something (i.e. illness, injury).

''2. How is it connected to your major?''

I'm a business major, but I'm pre-med so it's interesting to think about point charges/electric fields in the macroscopic lens and how they can combine with other forces (magnetic --> electromagnetic forces) to affect your internal systems and processes that occur without you consciously thinking about them.

''3. Is there an interesting industrial application?''

PEMF stands for pulsed electricmagnetic field, which is used by athletes during recovery or physical therapy. There is a difference in cell voltage between healthy cells and diseased cells. While healthy cells maintain a voltage around 70-100 mV, cells with illness have diminished voltage around 40mV. PEMF restores that optimal voltage in damaged cells. Through the utilization of low frequency pulsed electromagnetic fields at a high intensity, voltage in the damaged cells is increased. This stimulates cellular repair and recharges the body’s cells to optimize their performance.

PEMF therapy creates a magnetic field, which increases the movement of ions and electrolytes in the tissues and fluids of the body. The magnetic field helps cells increase ATP production, which restores and / or maintains normal cellular function, speeding up the tissues healing process, repairing damaged tissue, improving circulation, and increasing cellular energy.

==History==
[[File:CoulombCharles300px.jpg]]
''Charles de Coulomb''

Charles de Coulomb was born in June 14, 1736 in central France. He spent much of his early life in the military and was placed in regions throughout the world. He only began to do scientific experiments out of curiously on his military expeditions. However, when controversy arrived with him and the French bureaucracy coupled with the French Revolution, Coulomb had to leave France and thus really began his scientific career.

Between 1785 and 1791, de Coulomb wrote several key papers centered around multiple relations of electricity and magnetism. This helped him develop the principle known as Coulomb's Law, which confirmed that the force between two electrical charges is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. This is the same relationship that is seen in the electric field equation of a point charge.

== See also ==

[[Electric Field]] <br>
[[Electric Force]] <br>
[[Superposition Principle]] <br>
[[Electric Dipole]]

===Further reading===

Principles of Electrodynamics by Melvin Schwartz
ISBN: 9780486134673

Electricity and Magnetism: Edition 3 , Edward M. Purcell David J. Morin

===External links===

Some more information:

*http://hyperphysics.phy-astr.gsu.edu/hbase/electric/epoint.html
*http://www.physics.umd.edu/courses/Phys260/agashe/S10/notes/lecture18.pdf
*https://www.reliantphysicaltherapy.com/services/pulsed-electromagnetic-field-pemf

==References==

Chabay. (2000-2018). ''Matter & Interactions'' (4th ed.). John Wiley & Sons.

PY106 Notes. (n.d.). Retrieved November 27, 2016, from http://physics.bu.edu/~duffy/py106.html

Retrieved November 28, 2016, from http://www.biography.com/people/charles-de-coulomb-9259075#controversy-and-absolution

[[Category:Fields]]

Vectors

2020-10-06T04:37:50Z

Amummareddy3: /* Claimed by: Ashita Mummareddy (Fall 2020) */

== Claimed by: ==

This page defines and describes vectors.

==The Main Idea==

In mathematics and physics, a vector is a quantity with both a magnitude and a direction in space. The magnitude of a vector is a scalar value which can represent a variety of characteristics, depending on the situation. The magnitude of the vector (and the vector itself) has units corresponding to the characteristic it represents. The direction refers to the angle from the positive x -axis. Thus, the magnitude refers to the length and the direction refers to the angle. Note that a vector does not necessarily extend from one point in real, physical space to another; unless the magnitude is in units of length, the magnitude of the vector usually represents a property that exists at a single point in real, physical space, or at no position in particular.

===A Mathematical Model===

====Simple Examples of Vector Quantities====

To understand what it means for a vector to have both a magnitude and a direction, and to understand why the magnitude and direction together are often considered a single quantity, it can be helpful to consider an example. The [[Velocity]] of an object is an example of a vector quantity. The speed of the object, often given in meters per second (m/s), is a scalar value describing how quickly the object is moving. Speed is the magnitude of the velocity vector. However, the object's movement happens in a particular spatial direction, which the speed alone does not tell us. The direction of the object's movement is also a part of the velocity vector. Together, speed and direction comprise the velocity vector and give a complete description of an object's motion at a point in time. Another example of a vector is position; the distance of a point from the origin of a coordinate system can be represented as the magnitude of a vector, and this distance together with a direction describes exactly where a point can be found.

====Beginning Concepts====

A vector is typically written as a letter with an arrow over it. For example: <math>\vec{a} = \langle 1, 2, 3 \rangle</math>. The arrow may one-sided to make it easier to write. Alternatively, a vector might simply be written as a boldface letter: <b>a</b>. Which letter is used depends on the context; for example, <math>\vec{v}</math> represents velocity.

A specific component of a vector (see the section titled "forms") is denoted by a subscript: c<sub>x</sub>. For example, q<sub>y</sub> represents the y component of some vector <math>\vec{q}</math>.

Surrounding a vector by | symbols denotes its magnitude: <math>|\vec{b}|</math>. Alternatively, there are exists the notation <math>\overline{v}</math> and <math>\lVert \vec{v} \rVert_2</math>, with the latter denoting the Euclidean norm, which is the type of vector norm that you will see most often.

The magnitude of some vector is defined to be <math>\lVert \vec{v} \rVert_2 = \sqrt{v_x^2 + v_y^2 + v_z^2}</math>, where the axes of the 3D-map are mathematically notated as <math>\hat{x}, \hat{y}, \hat{z}</math>.

A unit vector is a vector whose magnitude is equal to one, and normalization is the process of setting a vector's magnitude to 1. A unit vector is denoted by a letter with a ^ symbol (called a "hat") written over it: <math>\hat{d}</math> (read as "d-hat"). Certain letters represent specific unit vectors. For example, <math>\hat{i}</math>, <math>\hat{j}</math>, and <math>\hat{k}</math> (alternatively, <math>\hat{x}</math>, <math>\hat{y}</math>, and <math>\hat{z}</math>) are unit vectors pointing in the +x, +y, and +z directions respectively. Non-Cartesian coordinate systems often have their own unit vectors; for example, 2D polar coordinates make use of the <math>\hat{r}</math> and <math>\hat{\theta}</math> unit vectors.

Unit vectors are defined to be for an arbitrary vector: <math>\hat{v} = \frac{\vec{v}}{\lVert \vec{v} \rVert_2}</math>

====Visually Representing Vectors====

Vectors are visually represented by arrows. The length of the arrow represents the magnitude of the vector, while the direction the arrow points in represents the direction of the vector. If a vector exists at a particular point in space, the "tail" of the arrow (the end without the V shape) should be placed at that point.

This example shows a visual representation of the velocity vector of a ball, which is moving to the right at a speed of 5m/s.

[[File:Vectorvisualrepresentation.png]]

====Important Vector Operations====

It is possible to perform a variety of mathematical operations on vectors, both with other vectors and with scalars. These operations appear in a variety of formulas in physics. To make the operations easier to learn, they are defined below assuming all vectors to be 3-dimensional; the more general n-dimensional definitions look more confusing. If necessary, it is easy to guess how to perform each operation with n-dimensional vectors by extrapolating from the 3-dimensional case.

<u>Addition:</u>

<math>\vec{a} + \vec{b} = \langle (a_x + b_x), (a_y + b_y), (a_z + b_z) \rangle</math>.

In other words, to add two vectors, simply add their like components to form the new components. Visually, if the tail of one vector is placed at the tip of another, their sum will extend from the tail of the second vector to the tip of the first:

[[File:Vectoraddition.png|600px]]

<u>Subtraction:</u>

<math>\vec{a} - \vec{b} = \langle (a_x - b_x), (a_y - b_y), (a_z - b_z)</math>.

In other words, to subtract two vectors, simply subtract their like components to form the new components.

<u>Multiplication by scalar:</u>

<math>k \vec{a} = \langle k \cdot a_x, k \cdot a_y, k \cdot a_z \rangle</math>.

In other words, multiplying a vector by a scalar multiplies each of that vector's components by that scalar. Note that this only affects a vector's magnitude, not its direction, unless the scalar is negative, in which case the direction of the vector is reversed.

[[File:Vectorscalarmultiplication.png|400px]]

<u>Division by scalar:</u>

Division by a scalar behaves exactly like scalar multiplication.

<math>\frac{\vec{a}}{k} = \langle \frac{a_x}{k}, \frac{a_y}{k}, \frac{a_z}{k} \rangle</math>.

<u>Dot product (also called scalar product):</u>

<math>\vec{a}\cdot\vec{b} =(a_x \cdot b_x) + (a_y \cdot b_y) + (a_z \cdot b_z)</math>.

In other words, the dot product of two vectors is the sum of the products of their like components. Note that this is a scalar value.

It is important to note that the dot product of two vectors has a specific value: <math>\vec{a}\cdot\vec{b} = |\vec{a}| \cdot |\vec{b}| \cdot \cos(\theta)</math>, where <math>\theta</math> is the angle between the vectors.

<u>Cross product (also called vector product):</u>

The simplest definition, which is also the one found on the formula sheet, is:

<math>\vec{a} \times \vec{b} = = \langle (a_yb_z - a_zb_y), (a_zb_x - a_xb_z), (a_xb_y - a_yb_x) \rangle = (a_yb_z - a_zb_y)\hat{i} + (a_zb_x - a_xb_z)\hat{j} + (a_xb_y - a_yb_x)\hat{k}</math>.

This is equivalent to the following matrix determinant, which may be easier to remember:

<math>\begin{vmatrix}
\hat{i} & \hat{j} & \hat{k} \\
a_x & a_y & a_z \\
b_x & b_y & b_z
\end{vmatrix}</math>

Note that this is a vector quantity. It is important to note that the <i>magnitude</i> of the cross product of two vectors has a specific value: <math>|\vec{a}\times\vec{b}| = |\vec{a}| \cdot |\vec{b}| \cdot \sin(\theta)</math>, where <math>\theta</math> is the angle between the vectors.

The direction of the cross product of two vectors is perpendicular to the plane in which those vectors lie and is given by the [[Right Hand Rule]]. 2D vectors do not have cross products. While the other operations listed here are commutative, associative (where the associative property is defined), and distributive over addition; cross product multiplication is not associative and is anticommutative (<math>\vec{a}\times\vec{b} = -\vec{b}\times\vec{a}</math>), meaning that if the order of the factors is reversed, their cross product will be reversed in direction.

<b>For more mathematically-advanced students</b>, I will invoke some higher-level linear algebra. Recall that the null space, defined as <math>Null(A) = \{ \vec{v} \in \mathbb{R}^n \mid A\vec{v} = \vec{0} \}</math> for an arbitrary matrix <math>A \in \mathbb{R}^{n \times n}</math>, is orthogonal, or normal, to its row space. That is, for all <math>\vec{v} \in Null(A)</math> and for all columns <math>a_{j:}</math>, we have <math>\sum^n_{i=1} a_{i,j} v_i = 0</math>.

For two vectors <math>\vec{a},\vec{b} \in \mathbb{R}^{3 \times 1}</math> (<i>note that this is only valid for 3x1 vectors</i>), if we think of <math>\vec{a}</math> as corresponding to a row in a matrix and <math>\vec{b}</math> to another, then the null space is equivalent to their cross-product. <b>This is why the right-hand rule works.</b> If we visualize the plane spanned by two vectors with our index and middle fingers, then their null space will be perpendicular to the intersection point of the two vectors, or rather the thumb. Likewise, if we use all of our fingers and our thumb to make an L-shape representing the span of the two vectors, upon curling our fingers, we will have the direction of the cross product, which is orthogonal to the two vectors represented by your thumb and fingers.

====Forms====

The information necessary to describe a specific vector can be presented in several forms.

<u>Magnitude and direction form</u>

In this form, the magnitude and the direction of the vector are explicitly stated. The statement describing direction might be a cardinal direction (ex. "north"), a direction on a graph (ex. "the +x direction"), or an angle (ex. "210<math>^\circ</math> from the x axis counterclockwise"), depending on the situation. Magnitude and direction form is often used in word problems because it is easy for humans to understand.

<u>Component form</u>

In this form, the vector is divided into components, each representing a different coordinate direction. In 2D space, these are the x and y directions. In 3D space, these are the x, y, and z directions. Each component tells how much the vector extends in that particular direction. Often, the three components are written enclosed by angle brackets and separated by commas. For example, the vector <2,0,-3> describes a vector that extends 2 units in the +x direction, 0 units in the y direction, and 3 units in the -z direction. Most vector operations described below can only be performed if the vectors are in component form, so this form may be necessary to do math for certain problems. Furthermore, programming languages store vectors in component form.

<u>Unit vector form</u>

In this form, the vector is expressed as a sum of unit vectors, each corresponding to a different coordinate direction. The symbols <math>\hat{i}</math>, <math>\hat{j}</math>, and <math>\hat{k}</math> OR the symbols <math>\hat{x}</math>, <math>\hat{y}</math>, and <math>\hat{z}</math> are used to represent unit vectors in the x, y, and z directions respectively. Consider the vector <2,0,-3>. It can be expressed in unit vector form as <math>2\hat{i} - 3\hat{k}</math>, meaning 2 times the x direction unit vector minus 3 times the z direction unit vector (see vector operations). While often considered its own form, unit vector form is very similar to component form, as the information describing the vector is stored in the same values. All references to component form in the rest of this page also apply to unit vector form.

[[File:Vectorsdifferentforms.png]]

Note that regardless of which form is used, an n-dimensional vector requires n values to mathematically describe. For example, consider a 3-dimensional vector. Describing this vector in component form requires an x value, a y value, and a z value. Describing it in magnitude and direction form requires one value to give the magnitude of the vector and two to give the its direction (the direction of a 3D vector could be described using, say, its polar angle <math>\theta</math> and its azimuthal angle <math>\phi</math>). A 1-dimensional vector (such as the velocity of a particle whose movement is constrained to the x axis) can be described using only 1 value whose sign indicates the vector's direction.

<u>Converting between forms</u>

It is possible to convert vectors from one form to another using simple trigonometry.

To find the magnitude of a vector in component form, use the Pythagorean theorem: add the squares of the components and take the square root of the result. For a 2D vector, <math>|\vec{a}| = \sqrt{a_x^2 + a_y^2}</math>.

To find the direction of a vector in component form, use inverse trigonometric functions. For a 2D vector, <math>\theta = \tan^{-1}(\frac{a_y}{a_x})</math>, where <math>\theta</math> is the angle vector <math>\vec{a}</math> makes with the x axis in the counterclockwise direction.

To find the components of a vector in magnitude and direction form, use trigonometric functions. For a 2D vector, <math>a_x = |\vec{a}|\cos\theta</math> and <math>a_y = |\vec{a}|\sin\theta</math>, where <math>\theta</math> is the angle vector <math>\vec{a}</math> makes with the x axis in the counterclockwise direction.

===A Computational Model===

In VPython, vector objects are in component form; each one has an x, y, and z component. Recall that in VPython, using the default camera orientation, the +x axis points to the right, the +y axis points upwards, and the +z axis points out of the plane of the screen towards the viewer. The constructor for a vector object is the word "vec" or "vector," and it takes three arguments, which define its x, y and z components respectively. A line to create a vector called "velocity" might look like this:

velocity = vec(3,-1,2)

To access or modify a specific component of a vector object, its name should be followed by a period and an x, y, or z. For example, to change the x component of the above velocity vector from 3 to 5, the following line might be used:

velocity.x = 5

In VPython, vectors have many uses. The position of each object is defined as a vector; the position vector's tail lies at the origin and its head lies at the center of the object in question. Furthermore, the dimensions of a rectangular prism (a "box" object) are defined as a vector; the x component determines its width, the y component its height, and the z component its thickness.

To graphically represent a vector such as an electric field, "arrow" objects should be used. In addition to taking a position vector (which determines the position of the arrow's tail), arrow objects take an "axis" vector, which determines their size and shape. To represent a vector, simply make it the axis of an arrow object.

==Examples==

===Simple===

Vector <math>\vec{a}</math> is <2,4,2>. Vector <math>\vec{b}</math> is <-1,1,3>. What is the magnitude of the vector <math>\vec{a} - 2\vec{b}</math>?

Solution:

<math>\vec{a} - 2\vec{b} = <2,4,2> - 2 * <-1,1,3> </math>

<math> = <2,4,2> - <-2,2,6> </math>

<math> = <4,2,-4></math>

We are asked to find the magnitude of this vector, so let us use the Pythagorean theorem with its components:

<math>|<4,2,-4>| = \sqrt{4^2 + 2^2 + (-4)^2}</math>

<math> = \sqrt{16 + 4 + 16} </math>

<math> = \sqrt{36} </math>

<math> = 6 </math>

===Intermediate===

An airplane is travelling in still air at 240m/s in the direction 35<math>^\circ</math> south of west. A wind begins to blow; the wind has a speed of 80m/s in the direction 15<math>^\circ</math> east of north. What should be the new velocity of the plane relative to the air around it to maintain its original trajectory? You may give your answer in component form (+x is east, +y is north).

[[File:Vectorsplaneproblem.png]]

Solution:

The vector sum of the new velocity of the plane <math>\vec{v_{p,1}}</math> and the velocity of the wind <math>\vec{v_w}</math> should equal the original velocity of the plane <math>\vec{v_{p,0}}</math> (see [[Relative Velocity]]):

[[File:Vectorsplanesolution.png]]

<math> \vec{v_{p,1}} + \vec{v_w} = \vec{v_{p,0}} </math>

<math> \vec{v_{p,1}} = \vec{v_{p,0}} - \vec{v_w} </math>

Let us convert the given vectors to component form for easier subtraction. The +x and +y directions will be east and north respectively.

<math>\vec{v_{p,0}} = <240\cos(215^\circ), 240\sin(215^\circ)></math>m/s (35<math>^\circ</math> south of west is 215<math>^\circ</math> above the x axis)

<math>\vec{v_{p,0}} = <-196.6, -137.7> </math>m/s

<math>\vec{v_w} = <80\cos(75^\circ), 80\sin(75^\circ)></math>m/s (15<math>^\circ</math> east of north is 75<math>^\circ</math> above the x axis)

<math>\vec{v_w} = <20.7, 77.3> </math>m/s

Now let us subtract:

<math> \vec{v_{p,1}} = \vec{v_{p,0}} - \vec{v_w} </math>

<math> \vec{v_{p,1}} = <-196.6, -137.7> - <20.7, 77.3> </math>m/s

<math> \vec{v_{p,1}} = <-217.3, -214.9> </math>m/s

===Difficult===

What is the angle between the vectors <2,5,-2> and <3,-4,-1>?

Solution:

The dot product between two vectors is equal to the product of their magnitudes times the cosine of the angle between them. Let us use this property to find the angle between the given vectors.

<math> <2,5,-2> \cdot <3,-4,-1> = |<2,5,-2>| * |<3,-4,-1>| * \cos\theta </math>

Rearranging this yields

<math> \theta = \cos^{-1}\frac{<2,5,-2> \cdot <3,-4,-1>}{|<2,5,-2>| * |<3,-4,-1>|} </math>

Let us evaluate the dot product and simplify:

<math> \theta = \cos^{-1}\frac{2(3) + 5(-4) + (-2)(-1)}{\sqrt{2^2 + 5^2 + (-2)^2} * \sqrt{3^2 + (-4)^2 + (-1)^2}} </math>

<math> \theta = \cos^{-1}\frac{-12}{\sqrt{33 * 26}} </math>

<math> \theta = 114^\circ </math>

==Connectedness==

Vectors are used in many fields and levels of physics. Kinematics, for example, studies the relationships between the position vector and its time derivatives (velocity, acceleration). Force is also a vector quantity, as are certain system properties such as linear and angular momentum. Vector fields are commonly used in electricity and magnetism, as well as in fluid dynamics.

==History==

It is unknown who first developed the idea of vectors, but the oldest known reference to vectors is in the work <i>Mechanics</i> by Hero of Alexandria (first century AD), which described their addition. At this point, however, the idea of a vector was little more than a line segment with a specific orientation; they had a length extending from one point in physical space to another but were not used to represent anything else.

In the early 19th century, several mathematicians and physicists (including Caspar Wessel (1745-1818), Jean Robert Argand (1768-1822), Carl Friedrich Gauss (1777-1855)), and William Rowan Hamilton (1805-1865) used 2D vectors to represent complex numbers; one component would represent the real value and another would represent the imaginary value. Hamilton would also become the first to use the word "vector." August Ferdinand Möbius (1790-1868) contributed to vector math in his 1827 book <i>The Barycentric Calculus</i>, in which he developed the convention of labeling vectors with letters and defined the multiplication of a vector by a scalar. Hermann Grassmann (1809-1877) wrote in his 1844 book <i>Ausdehnungslehre</i> (German for "The Calculus of Extension") that vectors could exist in space of any number of dimensions and described much of what would become linear algebra, which makes ample use of vectors.

The modern language and conventions surrounding vectors come largely from notes created by J. Willard Gibbs (1839--1903), a professor at Yale University.

==See also==

===External links===
Mathematical Computations on Vectors: [http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf [http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf]

Computational Work with Vectors: [http://vpython.org/contents/docs/vector.html http://vpython.org/contents/docs/vector.html]

Basics of Vectors: [https://www.physics.uoguelph.ca/tutorials/vectors/vectors.html https://www.physics.uoguelph.ca/tutorials/vectors/vectors.html]

===Further Reading===

Vector Analysis by Josiah Willard Gibbs

Introduction to Matrices and Vectors by Jacob T. Schwartz

==References==
[https://www.mathsisfun.com/algebra/vectors.html https://www.mathsisfun.com/algebra/vectors.html]

[http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf ]

[http://mathinsight.org/vector_introduction http://mathinsight.org/vector_introduction]

[http://www.math.mcgill.ca/labute/courses/133f03/VectorHistory.html http://www.math.mcgill.ca/labute/courses/133f03/VectorHistory.html]

[[Category: Geometry]]

Point Charge

2020-10-06T04:36:16Z

Amummareddy3: /* The Main Idea */

This page is all about the [[Electric Field]] due to a Point Charge.

==The Main Idea (CLAIMED BY ASHITA MUMMAREDDY FALL 2020) ==
(Ch 13.1 in ''Matter & Interactions Vol. 2: Modern Mechanics, 4th Edition by R. Chabay & B. Sherwood'')

'''Point Charge/Particle''' - an object with a radius that is very small compared to the distance between it and any other objects of interest in the system. Since it is very small, the object can be treated as if all of its charge and mass are concentrated at a single "point".
*Electrons and Protons are always considered to be point particles unless stated otherwise

<u> 2 types of point charges: </u>
*Protons (e) --> positive point charges, ( q = 1.6e-19 Coulombs)
*Electrons (-e) --> negative point charges, (q = -1.6e-19 Coulombs)

''Like'' point charges ''attract'', ''opposite'' point charges ''repel''.
ex.<table border> <tr>
<th> Point Charges </th>
<th> Result </th>
<th>Diagram</th>
</tr>
<tr>
<td> 1 proton, 1 electron</td>
<td> Attract </td>
<td>[[File:Proton_electron_attraction.png]]</td>
</tr>
<tr>
<td>2 protons </td>
<td> Repel </td>
<td>[[File:Proton_repulsion.png]]</td>
</tr>
<tr>
<td>2 electrons </td>
<td> Repel </td>
<td>[[File:Electron_repulsion.png]]</td>
</tr>
</table border>

===The Electric Field===
(Ch 13.3 in ''Matter & Interactions Vol. 2: Modern Mechanics, 4th Edition by R. Chabay & B. Sherwood'')

The electric field created by a charge is present throughout space at all times, whether or not there is another charge around to feel its effects.

Electric Field of a Charge Observed at a location: F = Eq
*F = Force on particle 2
*E = electric field at source location
*q = charge of particle 2
The magnitude of the electric field decreases with increasing distance from the point charge.

<table border>
<tr>
<td>The electric field of a positive point charge points radially outward</td>
<td>The electric field of a negative point charge points radially inward</td>
</tr>
<tr>
<td>[[File:Proton_electric_field.png]] </td>
<td> [[File:Electron_electric_field.png]] </td>
</tr>
</table border>

===A Mathematical Model===

====Electric Field due to Point Charge====
(Ch 13.4 in ''Matter & Interactions Vol. 2: Modern Mechanics, 4th Edition by R. Chabay & B. Sherwood'')

Electric Field of a Point Charge (<math>\vec E</math>):

<math>\vec E=\frac{1}{4 \pi \epsilon_0 } \frac{q}{\mid\vec r\mid ^2} \hat r</math> (Newtons/Coulomb)

*<math>\frac{1}{4 \pi \epsilon_0 } </math> is Coulomb's Constant and is approximately <math>8.987*10^{9}\frac{N m^2}{C^2} </math>
*'''''q''''' is the charge of the particle
*'''''r''''' is the magnitude of the distance between the observation location and the source location
*<math>\hat r </math> is the unit vector in the direction of the distance from the source location to the observation point.

The direction of the electric field at the observation location depends on the both the direction of <math>\hat r </math> and the sign of the source charge.
*If the source charge is positive, the field points away from the source charge.
*If the source charge is negative, the field points toward the source charge.

====Coulomb Force Law for Point Charges====
(Ch 13.2 in ''Matter & Interactions Vol. 2: Modern Mechanics, 4th Edition by R. Chabay & B. Sherwood'')

<math>\mid\vec F\mid=\frac{1}{4 \pi \epsilon_0 } \frac{\mid Q_1Q_2 \mid}{r^2}</math>

Coulomb's law is one of the four fundamental physical interactions, and it describes the magnitude of the electric force between two point-charges.
<math>Q_1, Q_2</math>= The charge of two particles of interest

*<math>\mid\vec F\mid=\frac{1}{4 \pi \epsilon_0 }</math> = constant,
* <math>Q_1, Q_2</math> = the magnitudes of the point charges
*r = The distance between the two particles

====Connection Between Electric Field and Force====
The force on a source charge is determined by <math> F = Eq </math> where '''''E''''' is the electric field and '''''q''''' is the charge of a test charge in Coulombs.

By solving for the electric field in <math> F = Eq </math>, with F modeled by Coulomb's Law, you get the equation for the electric field of the point charge:

<math> E = \frac{F}{q_2} = \frac{1}{4 \pi \epsilon_0 } \frac{q_1q_2}{r^2}\frac{1}{q_2}\hat r = \frac{1}{4 \pi \epsilon_0 } \frac{q_1}{r^2} \hat r </math>

===A Computational Model===

Below is a link to a code which can help visualize the Electric Field at various observation locations due to a proton. Notice how the arrows decrease in size by a factor of <math> \frac{1}{r^{2}} </math> as the observation location gets farther from the proton. The magnitude of the electric field decreases as the distance to the observation location increases.

[[File:First code.gif]]

Two adjacent point charges of opposite sign exhibit an electric field pattern that is characteristic of a dipole. This interaction is displayed in the code below. Notice how the electric field points towards the negatively charged point charge (blue) and away from the positively charged point charge (red).

[[File:Code_2.png]]

==Examples==

===Simple===
There is an electron at the origin. Calculate the electric field at <4, -3, 1> m.

<table border>
<tr>
<td> <b>Step 1.</b> Find <math>\hat r</math>
Find <math>\vec r_{obs} - \vec r_{electron}:

((4,-3,1) - (0,0,0) = <4,-3,1> </math>m.

Calculate the magnitude of <math>\vec r</math>:

(<math>\sqrt{4^2+(-3)^2+1^2}=\sqrt{26}</math> From <math>\vec r</math>, find the unit vector <math>\hat{r}.</math> <math> <\frac{4}{\sqrt{26}},\frac{-3}{\sqrt{26}},\frac{1}{\sqrt{26}}> </math>
</td></tr>

<tr><td>'''Step 2:''' Find the magnitude of the Electric Field

<math> E_{mag}= \frac{1}{4 \pi \epsilon_0 } \frac{q}{r^2} = \frac{1}{4 \pi \epsilon_0 } \frac{-1.6 * 10^{-19}}{26} </math> </td>
<td> </td></tr>

<td>'''Step 3:''' Multiply the magnitude of the Electric Field by <math>\hat{r}</math> to find the Electric Field
<math>E = \frac{1}{4 \pi \epsilon_0 } \frac{-1.6 * 10^{-19}}{26}*<\frac{4}{\sqrt{26}},\frac{-3}{\sqrt{26}},\frac{1}{\sqrt{26}}>

= <-4.34*10^{-11},3.26*10^{-11},-1.09*10^{-11}> N/C </math></td>
</table border>

===Middling===
A particle of unknown charge is located at <-0.21, 0.02, 0.11> m. Its electric field at point <-0.02, 0.31, 0.28> m is <math><0.124, 0.188, 0.109> </math> N/C. Find the magnitude and sign of the particle's charge.

Given both an observation location and a source location, one can find both r and <math>\hat{r}</math> Given the value of the electric field, one can also find the magnitude of the electric field. Then, using the equation for the magnitude of electric field of a point charge,<math> E_{mag}= \frac{1}{4 \pi \epsilon_0 } \frac{q}{r^2} </math> one can find the magnitude and sign of the charge.

<table border>
<tr>
<td> <b>Step 1.</b> Find <math>\vec r_{obs} - \vec r_{particle} </math>:

<math>\vec r = <-0.02, 0.31, 0.28> m - <-0.21, 0.02, 0.11> m = <0.19,0.29,0.17> m </math>

To find <math>\vec r_{mag} </math>, find the magnitude of <math><0.19,0.29,0.17></math>

<math>\sqrt{0.19^2+0.29^2+0.17^2}=\sqrt{0.1491}= 0.39</math>
</td></tr>

<tr><td><b>Step 2:</b> Find the magnitude of the Electric Field:

<math>E= <0.124, 0.188, 0.109> N/C</math>

<math>E_{mag} = (\sqrt{0.124^2+0.188^2+0.109^2}=\sqrt{0.0626}=0.25</math> </td></tr>

<td>'''Step 3:''' Find '''''q''''' by rearranging the equation for <math>E_{mag}</math>

<math> E_{mag}= \frac{1}{4 \pi \epsilon_0 } \frac{q}{r^2} </math>

By rearranging this equation we get

<math> q= {4 pi * ε_0 } *{r^2}*E_{mag} </math>

<math> q= {1/(9*10^9)} *{0.39^2}*0.25 </math>

<math> q= + 4.3*10^{-12} C </math></td>

</table border>

===Difficult===
The electric force on a -2mC particle at a location (3.98 , 3.98 , 3.98) m due to a particle at the origin is <math>< -5.5*10^{3} , -5.5*10^{3}, -5.5*10^{3}></math> N. What is the charge on the particle at the origin?

Given the force and charge on the particle, one can calculate the surrounding electric field. With this variable found, this problem becomes much like the last one.
<math> E_{mag}= \frac{1}{4 \pi \epsilon_0 } \frac{q}{r_{mag}^2} </math> to find the rmag value. To find <math>\hat r</math> we can find the direction of the electric field as that is obviously going to be in the same direction as <math>\hat r</math>. Then, once we find <math>\hat r</math>, all that is left to do is multiply <math>\hat r</math> by rmag and that will give us the <math> r</math> vector. We can then find the location of the particle as we know <math>r=r_{observation}-r_{particle}</math>

<table border>
<tr>
<td> <b>Step 1.</b> Find the magnitude of the Electric field:
<math> F = Eq </math>
<math> <5.5e3, -7.6e3, 0> = E * -2mC </math>
<math> E = \frac{< -5.5e3 , -5.5e3, -5.5e3>}{-2mC}

= <2.75e6 , 2.75e6, 2.75e6> </math> N/C
</td></tr>

<tr><td><b>Step 2:</b> Find <math>\vec r_{obs} - \vec r_{particle} </math>.

<math>\vec r = <3.98 , 3.98 , 3.98> m - <0 , 0 , 0> m = <3.98 , 3.98 , 3.98> m </math>

To find <math>\vec r_{mag} </math>, find the magnitude of <math><3.98 , 3.98 , 3.98></math>

<math>\sqrt{3.98^2+3.98^2+3.98^2}=\sqrt{47.52}= 6.9</math>
</td></tr>

<td> '''Step 4:''' Find '''''q''''' by rearranging the equation for <math>E_{mag}</math>

<math> E_{mag}= \frac{1}{4 \pi \epsilon_0 } \frac{q}{r^2} </math>

By rearranging this equation we get

<math> q= {4 pi * ε_0 } *{r^2}*E_{mag} </math>

<math> q= {1/(9e9)} *{6.9^{2}}*4.76e6} </math>

<math> q= + 0.253 C </math></td>

</table border>

==Connectedness==
''1. How is this topic connected to something that you are interested in?''

I think the topic is interesting because electric fields are inside the human body and we self-create these fields constantly. It's really cool to think about how your body might be responding and creating these fields continuously and what happens when the balances are thrown off by something (i.e. illness, injury).

''2. How is it connected to your major?''

I'm a business major, but I'm pre-med so it's interesting to think about point charges/electric fields in the macroscopic lens and how they can combine with other forces (magnetic --> electromagnetic forces) to affect your internal systems and processes that occur without you consciously thinking about them.

''3. Is there an interesting industrial application?''

PEMF stands for pulsed electricmagnetic field, which is used by athletes during recovery or physical therapy. There is a difference in cell voltage between healthy cells and diseased cells. While healthy cells maintain a voltage around 70-100 mV, cells with illness have diminished voltage around 40mV. PEMF restores that optimal voltage in damaged cells. Through the utilization of low frequency pulsed electromagnetic fields at a high intensity, voltage in the damaged cells is increased. This stimulates cellular repair and recharges the body’s cells to optimize their performance.

PEMF therapy creates a magnetic field, which increases the movement of ions and electrolytes in the tissues and fluids of the body. The magnetic field helps cells increase ATP production, which restores and / or maintains normal cellular function, speeding up the tissues healing process, repairing damaged tissue, improving circulation, and increasing cellular energy.

==History==
[[File:CoulombCharles300px.jpg]]
''Charles de Coulomb''

Charles de Coulomb was born in June 14, 1736 in central France. He spent much of his early life in the military and was placed in regions throughout the world. He only began to do scientific experiments out of curiously on his military expeditions. However, when controversy arrived with him and the French bureaucracy coupled with the French Revolution, Coulomb had to leave France and thus really began his scientific career.

Between 1785 and 1791, de Coulomb wrote several key papers centered around multiple relations of electricity and magnetism. This helped him develop the principle known as Coulomb's Law, which confirmed that the force between two electrical charges is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. This is the same relationship that is seen in the electric field equation of a point charge.

== See also ==

[[Electric Field]] <br>
[[Electric Force]] <br>
[[Superposition Principle]] <br>
[[Electric Dipole]]

===Further reading===

Principles of Electrodynamics by Melvin Schwartz
ISBN: 9780486134673

Electricity and Magnetism: Edition 3 , Edward M. Purcell David J. Morin

===External links===

Some more information:

*http://hyperphysics.phy-astr.gsu.edu/hbase/electric/epoint.html
*http://www.physics.umd.edu/courses/Phys260/agashe/S10/notes/lecture18.pdf
*https://www.reliantphysicaltherapy.com/services/pulsed-electromagnetic-field-pemf

==References==

Chabay. (2000-2018). ''Matter & Interactions'' (4th ed.). John Wiley & Sons.

PY106 Notes. (n.d.). Retrieved November 27, 2016, from http://physics.bu.edu/~duffy/py106.html

Retrieved November 28, 2016, from http://www.biography.com/people/charles-de-coulomb-9259075#controversy-and-absolution

[[Category:Fields]]

Point Charge

2020-10-06T04:35:38Z

Amummareddy3: Undo revision 38824 by Amummareddy3 (talk)

This page is all about the [[Electric Field]] due to a Point Charge.

==The Main Idea==
(Ch 13.1 in ''Matter & Interactions Vol. 2: Modern Mechanics, 4th Edition by R. Chabay & B. Sherwood'')

'''Point Charge/Particle''' - an object with a radius that is very small compared to the distance between it and any other objects of interest in the system. Since it is very small, the object can be treated as if all of its charge and mass are concentrated at a single "point".
*Electrons and Protons are always considered to be point particles unless stated otherwise

<u> 2 types of point charges: </u>
*Protons (e) --> positive point charges, ( q = 1.6e-19 Coulombs)
*Electrons (-e) --> negative point charges, (q = -1.6e-19 Coulombs)

''Like'' point charges ''attract'', ''opposite'' point charges ''repel''.
ex.<table border> <tr>
<th> Point Charges </th>
<th> Result </th>
<th>Diagram</th>
</tr>
<tr>
<td> 1 proton, 1 electron</td>
<td> Attract </td>
<td>[[File:Proton_electron_attraction.png]]</td>
</tr>
<tr>
<td>2 protons </td>
<td> Repel </td>
<td>[[File:Proton_repulsion.png]]</td>
</tr>
<tr>
<td>2 electrons </td>
<td> Repel </td>
<td>[[File:Electron_repulsion.png]]</td>
</tr>
</table border>

===The Electric Field===
(Ch 13.3 in ''Matter & Interactions Vol. 2: Modern Mechanics, 4th Edition by R. Chabay & B. Sherwood'')

The electric field created by a charge is present throughout space at all times, whether or not there is another charge around to feel its effects.

Electric Field of a Charge Observed at a location: F = Eq
*F = Force on particle 2
*E = electric field at source location
*q = charge of particle 2
The magnitude of the electric field decreases with increasing distance from the point charge.

<table border>
<tr>
<td>The electric field of a positive point charge points radially outward</td>
<td>The electric field of a negative point charge points radially inward</td>
</tr>
<tr>
<td>[[File:Proton_electric_field.png]] </td>
<td> [[File:Electron_electric_field.png]] </td>
</tr>
</table border>

===A Mathematical Model===

====Electric Field due to Point Charge====
(Ch 13.4 in ''Matter & Interactions Vol. 2: Modern Mechanics, 4th Edition by R. Chabay & B. Sherwood'')

Electric Field of a Point Charge (<math>\vec E</math>):

<math>\vec E=\frac{1}{4 \pi \epsilon_0 } \frac{q}{\mid\vec r\mid ^2} \hat r</math> (Newtons/Coulomb)

*<math>\frac{1}{4 \pi \epsilon_0 } </math> is Coulomb's Constant and is approximately <math>8.987*10^{9}\frac{N m^2}{C^2} </math>
*'''''q''''' is the charge of the particle
*'''''r''''' is the magnitude of the distance between the observation location and the source location
*<math>\hat r </math> is the unit vector in the direction of the distance from the source location to the observation point.

The direction of the electric field at the observation location depends on the both the direction of <math>\hat r </math> and the sign of the source charge.
*If the source charge is positive, the field points away from the source charge.
*If the source charge is negative, the field points toward the source charge.

====Coulomb Force Law for Point Charges====
(Ch 13.2 in ''Matter & Interactions Vol. 2: Modern Mechanics, 4th Edition by R. Chabay & B. Sherwood'')

<math>\mid\vec F\mid=\frac{1}{4 \pi \epsilon_0 } \frac{\mid Q_1Q_2 \mid}{r^2}</math>

Coulomb's law is one of the four fundamental physical interactions, and it describes the magnitude of the electric force between two point-charges.
<math>Q_1, Q_2</math>= The charge of two particles of interest

*<math>\mid\vec F\mid=\frac{1}{4 \pi \epsilon_0 }</math> = constant,
* <math>Q_1, Q_2</math> = the magnitudes of the point charges
*r = The distance between the two particles

====Connection Between Electric Field and Force====
The force on a source charge is determined by <math> F = Eq </math> where '''''E''''' is the electric field and '''''q''''' is the charge of a test charge in Coulombs.

By solving for the electric field in <math> F = Eq </math>, with F modeled by Coulomb's Law, you get the equation for the electric field of the point charge:

<math> E = \frac{F}{q_2} = \frac{1}{4 \pi \epsilon_0 } \frac{q_1q_2}{r^2}\frac{1}{q_2}\hat r = \frac{1}{4 \pi \epsilon_0 } \frac{q_1}{r^2} \hat r </math>

===A Computational Model===

Below is a link to a code which can help visualize the Electric Field at various observation locations due to a proton. Notice how the arrows decrease in size by a factor of <math> \frac{1}{r^{2}} </math> as the observation location gets farther from the proton. The magnitude of the electric field decreases as the distance to the observation location increases.

[[File:First code.gif]]

Two adjacent point charges of opposite sign exhibit an electric field pattern that is characteristic of a dipole. This interaction is displayed in the code below. Notice how the electric field points towards the negatively charged point charge (blue) and away from the positively charged point charge (red).

[[File:Code_2.png]]

==Examples==

===Simple===
There is an electron at the origin. Calculate the electric field at <4, -3, 1> m.

<table border>
<tr>
<td> <b>Step 1.</b> Find <math>\hat r</math>
Find <math>\vec r_{obs} - \vec r_{electron}:

((4,-3,1) - (0,0,0) = <4,-3,1> </math>m.

Calculate the magnitude of <math>\vec r</math>:

(<math>\sqrt{4^2+(-3)^2+1^2}=\sqrt{26}</math> From <math>\vec r</math>, find the unit vector <math>\hat{r}.</math> <math> <\frac{4}{\sqrt{26}},\frac{-3}{\sqrt{26}},\frac{1}{\sqrt{26}}> </math>
</td></tr>

<tr><td>'''Step 2:''' Find the magnitude of the Electric Field

<math> E_{mag}= \frac{1}{4 \pi \epsilon_0 } \frac{q}{r^2} = \frac{1}{4 \pi \epsilon_0 } \frac{-1.6 * 10^{-19}}{26} </math> </td>
<td> </td></tr>

<td>'''Step 3:''' Multiply the magnitude of the Electric Field by <math>\hat{r}</math> to find the Electric Field
<math>E = \frac{1}{4 \pi \epsilon_0 } \frac{-1.6 * 10^{-19}}{26}*<\frac{4}{\sqrt{26}},\frac{-3}{\sqrt{26}},\frac{1}{\sqrt{26}}>

= <-4.34*10^{-11},3.26*10^{-11},-1.09*10^{-11}> N/C </math></td>
</table border>

===Middling===
A particle of unknown charge is located at <-0.21, 0.02, 0.11> m. Its electric field at point <-0.02, 0.31, 0.28> m is <math><0.124, 0.188, 0.109> </math> N/C. Find the magnitude and sign of the particle's charge.

Given both an observation location and a source location, one can find both r and <math>\hat{r}</math> Given the value of the electric field, one can also find the magnitude of the electric field. Then, using the equation for the magnitude of electric field of a point charge,<math> E_{mag}= \frac{1}{4 \pi \epsilon_0 } \frac{q}{r^2} </math> one can find the magnitude and sign of the charge.

<table border>
<tr>
<td> <b>Step 1.</b> Find <math>\vec r_{obs} - \vec r_{particle} </math>:

<math>\vec r = <-0.02, 0.31, 0.28> m - <-0.21, 0.02, 0.11> m = <0.19,0.29,0.17> m </math>

To find <math>\vec r_{mag} </math>, find the magnitude of <math><0.19,0.29,0.17></math>

<math>\sqrt{0.19^2+0.29^2+0.17^2}=\sqrt{0.1491}= 0.39</math>
</td></tr>

<tr><td><b>Step 2:</b> Find the magnitude of the Electric Field:

<math>E= <0.124, 0.188, 0.109> N/C</math>

<math>E_{mag} = (\sqrt{0.124^2+0.188^2+0.109^2}=\sqrt{0.0626}=0.25</math> </td></tr>

<td>'''Step 3:''' Find '''''q''''' by rearranging the equation for <math>E_{mag}</math>

<math> E_{mag}= \frac{1}{4 \pi \epsilon_0 } \frac{q}{r^2} </math>

By rearranging this equation we get

<math> q= {4 pi * ε_0 } *{r^2}*E_{mag} </math>

<math> q= {1/(9*10^9)} *{0.39^2}*0.25 </math>

<math> q= + 4.3*10^{-12} C </math></td>

</table border>

===Difficult===
The electric force on a -2mC particle at a location (3.98 , 3.98 , 3.98) m due to a particle at the origin is <math>< -5.5*10^{3} , -5.5*10^{3}, -5.5*10^{3}></math> N. What is the charge on the particle at the origin?

Given the force and charge on the particle, one can calculate the surrounding electric field. With this variable found, this problem becomes much like the last one.
<math> E_{mag}= \frac{1}{4 \pi \epsilon_0 } \frac{q}{r_{mag}^2} </math> to find the rmag value. To find <math>\hat r</math> we can find the direction of the electric field as that is obviously going to be in the same direction as <math>\hat r</math>. Then, once we find <math>\hat r</math>, all that is left to do is multiply <math>\hat r</math> by rmag and that will give us the <math> r</math> vector. We can then find the location of the particle as we know <math>r=r_{observation}-r_{particle}</math>

<table border>
<tr>
<td> <b>Step 1.</b> Find the magnitude of the Electric field:
<math> F = Eq </math>
<math> <5.5e3, -7.6e3, 0> = E * -2mC </math>
<math> E = \frac{< -5.5e3 , -5.5e3, -5.5e3>}{-2mC}

= <2.75e6 , 2.75e6, 2.75e6> </math> N/C
</td></tr>

<tr><td><b>Step 2:</b> Find <math>\vec r_{obs} - \vec r_{particle} </math>.

<math>\vec r = <3.98 , 3.98 , 3.98> m - <0 , 0 , 0> m = <3.98 , 3.98 , 3.98> m </math>

To find <math>\vec r_{mag} </math>, find the magnitude of <math><3.98 , 3.98 , 3.98></math>

<math>\sqrt{3.98^2+3.98^2+3.98^2}=\sqrt{47.52}= 6.9</math>
</td></tr>

<td> '''Step 4:''' Find '''''q''''' by rearranging the equation for <math>E_{mag}</math>

<math> E_{mag}= \frac{1}{4 \pi \epsilon_0 } \frac{q}{r^2} </math>

By rearranging this equation we get

<math> q= {4 pi * ε_0 } *{r^2}*E_{mag} </math>

<math> q= {1/(9e9)} *{6.9^{2}}*4.76e6} </math>

<math> q= + 0.253 C </math></td>

</table border>

==Connectedness==
''1. How is this topic connected to something that you are interested in?''

I think the topic is interesting because electric fields are inside the human body and we self-create these fields constantly. It's really cool to think about how your body might be responding and creating these fields continuously and what happens when the balances are thrown off by something (i.e. illness, injury).

''2. How is it connected to your major?''

I'm a business major, but I'm pre-med so it's interesting to think about point charges/electric fields in the macroscopic lens and how they can combine with other forces (magnetic --> electromagnetic forces) to affect your internal systems and processes that occur without you consciously thinking about them.

''3. Is there an interesting industrial application?''

PEMF stands for pulsed electricmagnetic field, which is used by athletes during recovery or physical therapy. There is a difference in cell voltage between healthy cells and diseased cells. While healthy cells maintain a voltage around 70-100 mV, cells with illness have diminished voltage around 40mV. PEMF restores that optimal voltage in damaged cells. Through the utilization of low frequency pulsed electromagnetic fields at a high intensity, voltage in the damaged cells is increased. This stimulates cellular repair and recharges the body’s cells to optimize their performance.

PEMF therapy creates a magnetic field, which increases the movement of ions and electrolytes in the tissues and fluids of the body. The magnetic field helps cells increase ATP production, which restores and / or maintains normal cellular function, speeding up the tissues healing process, repairing damaged tissue, improving circulation, and increasing cellular energy.

==History==
[[File:CoulombCharles300px.jpg]]
''Charles de Coulomb''

Charles de Coulomb was born in June 14, 1736 in central France. He spent much of his early life in the military and was placed in regions throughout the world. He only began to do scientific experiments out of curiously on his military expeditions. However, when controversy arrived with him and the French bureaucracy coupled with the French Revolution, Coulomb had to leave France and thus really began his scientific career.

Between 1785 and 1791, de Coulomb wrote several key papers centered around multiple relations of electricity and magnetism. This helped him develop the principle known as Coulomb's Law, which confirmed that the force between two electrical charges is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. This is the same relationship that is seen in the electric field equation of a point charge.

== See also ==

[[Electric Field]] <br>
[[Electric Force]] <br>
[[Superposition Principle]] <br>
[[Electric Dipole]]

===Further reading===

Principles of Electrodynamics by Melvin Schwartz
ISBN: 9780486134673

Electricity and Magnetism: Edition 3 , Edward M. Purcell David J. Morin

===External links===

Some more information:

*http://hyperphysics.phy-astr.gsu.edu/hbase/electric/epoint.html
*http://www.physics.umd.edu/courses/Phys260/agashe/S10/notes/lecture18.pdf
*https://www.reliantphysicaltherapy.com/services/pulsed-electromagnetic-field-pemf

==References==

Chabay. (2000-2018). ''Matter & Interactions'' (4th ed.). John Wiley & Sons.

PY106 Notes. (n.d.). Retrieved November 27, 2016, from http://physics.bu.edu/~duffy/py106.html

Retrieved November 28, 2016, from http://www.biography.com/people/charles-de-coulomb-9259075#controversy-and-absolution

[[Category:Fields]]

Point Charge

2020-10-06T04:34:44Z

Amummareddy3: /* The Main Idea */

This page is all about the [[Electric Field]] due to a Point Charge.

==The Main Idea== CLAIMED BY ASHITA MUMMAREDDY (FALL 2020)
(Ch 13.1 in ''Matter & Interactions Vol. 2: Modern Mechanics, 4th Edition by R. Chabay & B. Sherwood'')

'''Point Charge/Particle''' - an object with a radius that is very small compared to the distance between it and any other objects of interest in the system. Since it is very small, the object can be treated as if all of its charge and mass are concentrated at a single "point".
*Electrons and Protons are always considered to be point particles unless stated otherwise

<u> 2 types of point charges: </u>
*Protons (e) --> positive point charges, ( q = 1.6e-19 Coulombs)
*Electrons (-e) --> negative point charges, (q = -1.6e-19 Coulombs)

''Like'' point charges ''attract'', ''opposite'' point charges ''repel''.
ex.<table border> <tr>
<th> Point Charges </th>
<th> Result </th>
<th>Diagram</th>
</tr>
<tr>
<td> 1 proton, 1 electron</td>
<td> Attract </td>
<td>[[File:Proton_electron_attraction.png]]</td>
</tr>
<tr>
<td>2 protons </td>
<td> Repel </td>
<td>[[File:Proton_repulsion.png]]</td>
</tr>
<tr>
<td>2 electrons </td>
<td> Repel </td>
<td>[[File:Electron_repulsion.png]]</td>
</tr>
</table border>

===The Electric Field===
(Ch 13.3 in ''Matter & Interactions Vol. 2: Modern Mechanics, 4th Edition by R. Chabay & B. Sherwood'')

The electric field created by a charge is present throughout space at all times, whether or not there is another charge around to feel its effects.

Electric Field of a Charge Observed at a location: F = Eq
*F = Force on particle 2
*E = electric field at source location
*q = charge of particle 2
The magnitude of the electric field decreases with increasing distance from the point charge.

<table border>
<tr>
<td>The electric field of a positive point charge points radially outward</td>
<td>The electric field of a negative point charge points radially inward</td>
</tr>
<tr>
<td>[[File:Proton_electric_field.png]] </td>
<td> [[File:Electron_electric_field.png]] </td>
</tr>
</table border>

===A Mathematical Model===

====Electric Field due to Point Charge====
(Ch 13.4 in ''Matter & Interactions Vol. 2: Modern Mechanics, 4th Edition by R. Chabay & B. Sherwood'')

Electric Field of a Point Charge (<math>\vec E</math>):

<math>\vec E=\frac{1}{4 \pi \epsilon_0 } \frac{q}{\mid\vec r\mid ^2} \hat r</math> (Newtons/Coulomb)

*<math>\frac{1}{4 \pi \epsilon_0 } </math> is Coulomb's Constant and is approximately <math>8.987*10^{9}\frac{N m^2}{C^2} </math>
*'''''q''''' is the charge of the particle
*'''''r''''' is the magnitude of the distance between the observation location and the source location
*<math>\hat r </math> is the unit vector in the direction of the distance from the source location to the observation point.

The direction of the electric field at the observation location depends on the both the direction of <math>\hat r </math> and the sign of the source charge.
*If the source charge is positive, the field points away from the source charge.
*If the source charge is negative, the field points toward the source charge.

====Coulomb Force Law for Point Charges====
(Ch 13.2 in ''Matter & Interactions Vol. 2: Modern Mechanics, 4th Edition by R. Chabay & B. Sherwood'')

<math>\mid\vec F\mid=\frac{1}{4 \pi \epsilon_0 } \frac{\mid Q_1Q_2 \mid}{r^2}</math>

Coulomb's law is one of the four fundamental physical interactions, and it describes the magnitude of the electric force between two point-charges.
<math>Q_1, Q_2</math>= The charge of two particles of interest

*<math>\mid\vec F\mid=\frac{1}{4 \pi \epsilon_0 }</math> = constant,
* <math>Q_1, Q_2</math> = the magnitudes of the point charges
*r = The distance between the two particles

====Connection Between Electric Field and Force====
The force on a source charge is determined by <math> F = Eq </math> where '''''E''''' is the electric field and '''''q''''' is the charge of a test charge in Coulombs.

By solving for the electric field in <math> F = Eq </math>, with F modeled by Coulomb's Law, you get the equation for the electric field of the point charge:

<math> E = \frac{F}{q_2} = \frac{1}{4 \pi \epsilon_0 } \frac{q_1q_2}{r^2}\frac{1}{q_2}\hat r = \frac{1}{4 \pi \epsilon_0 } \frac{q_1}{r^2} \hat r </math>

===A Computational Model===

Below is a link to a code which can help visualize the Electric Field at various observation locations due to a proton. Notice how the arrows decrease in size by a factor of <math> \frac{1}{r^{2}} </math> as the observation location gets farther from the proton. The magnitude of the electric field decreases as the distance to the observation location increases.

[[File:First code.gif]]

Two adjacent point charges of opposite sign exhibit an electric field pattern that is characteristic of a dipole. This interaction is displayed in the code below. Notice how the electric field points towards the negatively charged point charge (blue) and away from the positively charged point charge (red).

[[File:Code_2.png]]

==Examples==

===Simple===
There is an electron at the origin. Calculate the electric field at <4, -3, 1> m.

<table border>
<tr>
<td> <b>Step 1.</b> Find <math>\hat r</math>
Find <math>\vec r_{obs} - \vec r_{electron}:

((4,-3,1) - (0,0,0) = <4,-3,1> </math>m.

Calculate the magnitude of <math>\vec r</math>:

(<math>\sqrt{4^2+(-3)^2+1^2}=\sqrt{26}</math> From <math>\vec r</math>, find the unit vector <math>\hat{r}.</math> <math> <\frac{4}{\sqrt{26}},\frac{-3}{\sqrt{26}},\frac{1}{\sqrt{26}}> </math>
</td></tr>

<tr><td>'''Step 2:''' Find the magnitude of the Electric Field

<math> E_{mag}= \frac{1}{4 \pi \epsilon_0 } \frac{q}{r^2} = \frac{1}{4 \pi \epsilon_0 } \frac{-1.6 * 10^{-19}}{26} </math> </td>
<td> </td></tr>

<td>'''Step 3:''' Multiply the magnitude of the Electric Field by <math>\hat{r}</math> to find the Electric Field
<math>E = \frac{1}{4 \pi \epsilon_0 } \frac{-1.6 * 10^{-19}}{26}*<\frac{4}{\sqrt{26}},\frac{-3}{\sqrt{26}},\frac{1}{\sqrt{26}}>

= <-4.34*10^{-11},3.26*10^{-11},-1.09*10^{-11}> N/C </math></td>
</table border>

===Middling===
A particle of unknown charge is located at <-0.21, 0.02, 0.11> m. Its electric field at point <-0.02, 0.31, 0.28> m is <math><0.124, 0.188, 0.109> </math> N/C. Find the magnitude and sign of the particle's charge.

Given both an observation location and a source location, one can find both r and <math>\hat{r}</math> Given the value of the electric field, one can also find the magnitude of the electric field. Then, using the equation for the magnitude of electric field of a point charge,<math> E_{mag}= \frac{1}{4 \pi \epsilon_0 } \frac{q}{r^2} </math> one can find the magnitude and sign of the charge.

<table border>
<tr>
<td> <b>Step 1.</b> Find <math>\vec r_{obs} - \vec r_{particle} </math>:

<math>\vec r = <-0.02, 0.31, 0.28> m - <-0.21, 0.02, 0.11> m = <0.19,0.29,0.17> m </math>

To find <math>\vec r_{mag} </math>, find the magnitude of <math><0.19,0.29,0.17></math>

<math>\sqrt{0.19^2+0.29^2+0.17^2}=\sqrt{0.1491}= 0.39</math>
</td></tr>

<tr><td><b>Step 2:</b> Find the magnitude of the Electric Field:

<math>E= <0.124, 0.188, 0.109> N/C</math>

<math>E_{mag} = (\sqrt{0.124^2+0.188^2+0.109^2}=\sqrt{0.0626}=0.25</math> </td></tr>

<td>'''Step 3:''' Find '''''q''''' by rearranging the equation for <math>E_{mag}</math>

<math> E_{mag}= \frac{1}{4 \pi \epsilon_0 } \frac{q}{r^2} </math>

By rearranging this equation we get

<math> q= {4 pi * ε_0 } *{r^2}*E_{mag} </math>

<math> q= {1/(9*10^9)} *{0.39^2}*0.25 </math>

<math> q= + 4.3*10^{-12} C </math></td>

</table border>

===Difficult===
The electric force on a -2mC particle at a location (3.98 , 3.98 , 3.98) m due to a particle at the origin is <math>< -5.5*10^{3} , -5.5*10^{3}, -5.5*10^{3}></math> N. What is the charge on the particle at the origin?

Given the force and charge on the particle, one can calculate the surrounding electric field. With this variable found, this problem becomes much like the last one.
<math> E_{mag}= \frac{1}{4 \pi \epsilon_0 } \frac{q}{r_{mag}^2} </math> to find the rmag value. To find <math>\hat r</math> we can find the direction of the electric field as that is obviously going to be in the same direction as <math>\hat r</math>. Then, once we find <math>\hat r</math>, all that is left to do is multiply <math>\hat r</math> by rmag and that will give us the <math> r</math> vector. We can then find the location of the particle as we know <math>r=r_{observation}-r_{particle}</math>

<table border>
<tr>
<td> <b>Step 1.</b> Find the magnitude of the Electric field:
<math> F = Eq </math>
<math> <5.5e3, -7.6e3, 0> = E * -2mC </math>
<math> E = \frac{< -5.5e3 , -5.5e3, -5.5e3>}{-2mC}

= <2.75e6 , 2.75e6, 2.75e6> </math> N/C
</td></tr>

<tr><td><b>Step 2:</b> Find <math>\vec r_{obs} - \vec r_{particle} </math>.

<math>\vec r = <3.98 , 3.98 , 3.98> m - <0 , 0 , 0> m = <3.98 , 3.98 , 3.98> m </math>

To find <math>\vec r_{mag} </math>, find the magnitude of <math><3.98 , 3.98 , 3.98></math>

<math>\sqrt{3.98^2+3.98^2+3.98^2}=\sqrt{47.52}= 6.9</math>
</td></tr>

<td> '''Step 4:''' Find '''''q''''' by rearranging the equation for <math>E_{mag}</math>

<math> E_{mag}= \frac{1}{4 \pi \epsilon_0 } \frac{q}{r^2} </math>

By rearranging this equation we get

<math> q= {4 pi * ε_0 } *{r^2}*E_{mag} </math>

<math> q= {1/(9e9)} *{6.9^{2}}*4.76e6} </math>

<math> q= + 0.253 C </math></td>

</table border>

==Connectedness==
''1. How is this topic connected to something that you are interested in?''

I think the topic is interesting because electric fields are inside the human body and we self-create these fields constantly. It's really cool to think about how your body might be responding and creating these fields continuously and what happens when the balances are thrown off by something (i.e. illness, injury).

''2. How is it connected to your major?''

I'm a business major, but I'm pre-med so it's interesting to think about point charges/electric fields in the macroscopic lens and how they can combine with other forces (magnetic --> electromagnetic forces) to affect your internal systems and processes that occur without you consciously thinking about them.

''3. Is there an interesting industrial application?''

PEMF stands for pulsed electricmagnetic field, which is used by athletes during recovery or physical therapy. There is a difference in cell voltage between healthy cells and diseased cells. While healthy cells maintain a voltage around 70-100 mV, cells with illness have diminished voltage around 40mV. PEMF restores that optimal voltage in damaged cells. Through the utilization of low frequency pulsed electromagnetic fields at a high intensity, voltage in the damaged cells is increased. This stimulates cellular repair and recharges the body’s cells to optimize their performance.

PEMF therapy creates a magnetic field, which increases the movement of ions and electrolytes in the tissues and fluids of the body. The magnetic field helps cells increase ATP production, which restores and / or maintains normal cellular function, speeding up the tissues healing process, repairing damaged tissue, improving circulation, and increasing cellular energy.

==History==
[[File:CoulombCharles300px.jpg]]
''Charles de Coulomb''

Charles de Coulomb was born in June 14, 1736 in central France. He spent much of his early life in the military and was placed in regions throughout the world. He only began to do scientific experiments out of curiously on his military expeditions. However, when controversy arrived with him and the French bureaucracy coupled with the French Revolution, Coulomb had to leave France and thus really began his scientific career.

Between 1785 and 1791, de Coulomb wrote several key papers centered around multiple relations of electricity and magnetism. This helped him develop the principle known as Coulomb's Law, which confirmed that the force between two electrical charges is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. This is the same relationship that is seen in the electric field equation of a point charge.

== See also ==

[[Electric Field]] <br>
[[Electric Force]] <br>
[[Superposition Principle]] <br>
[[Electric Dipole]]

===Further reading===

Principles of Electrodynamics by Melvin Schwartz
ISBN: 9780486134673

Electricity and Magnetism: Edition 3 , Edward M. Purcell David J. Morin

===External links===

Some more information:

*http://hyperphysics.phy-astr.gsu.edu/hbase/electric/epoint.html
*http://www.physics.umd.edu/courses/Phys260/agashe/S10/notes/lecture18.pdf
*https://www.reliantphysicaltherapy.com/services/pulsed-electromagnetic-field-pemf

==References==

Chabay. (2000-2018). ''Matter & Interactions'' (4th ed.). John Wiley & Sons.

PY106 Notes. (n.d.). Retrieved November 27, 2016, from http://physics.bu.edu/~duffy/py106.html

Retrieved November 28, 2016, from http://www.biography.com/people/charles-de-coulomb-9259075#controversy-and-absolution

[[Category:Fields]]

Point Charge

2020-10-06T04:34:21Z

Amummareddy3:

This page is all about the [[Electric Field]] due to a Point Charge.

==The Main Idea==
(Ch 13.1 in ''Matter & Interactions Vol. 2: Modern Mechanics, 4th Edition by R. Chabay & B. Sherwood'')

'''Point Charge/Particle''' - an object with a radius that is very small compared to the distance between it and any other objects of interest in the system. Since it is very small, the object can be treated as if all of its charge and mass are concentrated at a single "point".
*Electrons and Protons are always considered to be point particles unless stated otherwise

<u> 2 types of point charges: </u>
*Protons (e) --> positive point charges, ( q = 1.6e-19 Coulombs)
*Electrons (-e) --> negative point charges, (q = -1.6e-19 Coulombs)

''Like'' point charges ''attract'', ''opposite'' point charges ''repel''.
ex.<table border> <tr>
<th> Point Charges </th>
<th> Result </th>
<th>Diagram</th>
</tr>
<tr>
<td> 1 proton, 1 electron</td>
<td> Attract </td>
<td>[[File:Proton_electron_attraction.png]]</td>
</tr>
<tr>
<td>2 protons </td>
<td> Repel </td>
<td>[[File:Proton_repulsion.png]]</td>
</tr>
<tr>
<td>2 electrons </td>
<td> Repel </td>
<td>[[File:Electron_repulsion.png]]</td>
</tr>
</table border>

===The Electric Field===
(Ch 13.3 in ''Matter & Interactions Vol. 2: Modern Mechanics, 4th Edition by R. Chabay & B. Sherwood'')

The electric field created by a charge is present throughout space at all times, whether or not there is another charge around to feel its effects.

Electric Field of a Charge Observed at a location: F = Eq
*F = Force on particle 2
*E = electric field at source location
*q = charge of particle 2
The magnitude of the electric field decreases with increasing distance from the point charge.

<table border>
<tr>
<td>The electric field of a positive point charge points radially outward</td>
<td>The electric field of a negative point charge points radially inward</td>
</tr>
<tr>
<td>[[File:Proton_electric_field.png]] </td>
<td> [[File:Electron_electric_field.png]] </td>
</tr>
</table border>

===A Mathematical Model===

====Electric Field due to Point Charge====
(Ch 13.4 in ''Matter & Interactions Vol. 2: Modern Mechanics, 4th Edition by R. Chabay & B. Sherwood'')

Electric Field of a Point Charge (<math>\vec E</math>):

<math>\vec E=\frac{1}{4 \pi \epsilon_0 } \frac{q}{\mid\vec r\mid ^2} \hat r</math> (Newtons/Coulomb)

*<math>\frac{1}{4 \pi \epsilon_0 } </math> is Coulomb's Constant and is approximately <math>8.987*10^{9}\frac{N m^2}{C^2} </math>
*'''''q''''' is the charge of the particle
*'''''r''''' is the magnitude of the distance between the observation location and the source location
*<math>\hat r </math> is the unit vector in the direction of the distance from the source location to the observation point.

The direction of the electric field at the observation location depends on the both the direction of <math>\hat r </math> and the sign of the source charge.
*If the source charge is positive, the field points away from the source charge.
*If the source charge is negative, the field points toward the source charge.

====Coulomb Force Law for Point Charges====
(Ch 13.2 in ''Matter & Interactions Vol. 2: Modern Mechanics, 4th Edition by R. Chabay & B. Sherwood'')

<math>\mid\vec F\mid=\frac{1}{4 \pi \epsilon_0 } \frac{\mid Q_1Q_2 \mid}{r^2}</math>

Coulomb's law is one of the four fundamental physical interactions, and it describes the magnitude of the electric force between two point-charges.
<math>Q_1, Q_2</math>= The charge of two particles of interest

*<math>\mid\vec F\mid=\frac{1}{4 \pi \epsilon_0 }</math> = constant,
* <math>Q_1, Q_2</math> = the magnitudes of the point charges
*r = The distance between the two particles

====Connection Between Electric Field and Force====
The force on a source charge is determined by <math> F = Eq </math> where '''''E''''' is the electric field and '''''q''''' is the charge of a test charge in Coulombs.

By solving for the electric field in <math> F = Eq </math>, with F modeled by Coulomb's Law, you get the equation for the electric field of the point charge:

<math> E = \frac{F}{q_2} = \frac{1}{4 \pi \epsilon_0 } \frac{q_1q_2}{r^2}\frac{1}{q_2}\hat r = \frac{1}{4 \pi \epsilon_0 } \frac{q_1}{r^2} \hat r </math>

===A Computational Model===

Below is a link to a code which can help visualize the Electric Field at various observation locations due to a proton. Notice how the arrows decrease in size by a factor of <math> \frac{1}{r^{2}} </math> as the observation location gets farther from the proton. The magnitude of the electric field decreases as the distance to the observation location increases.

[[File:First code.gif]]

Two adjacent point charges of opposite sign exhibit an electric field pattern that is characteristic of a dipole. This interaction is displayed in the code below. Notice how the electric field points towards the negatively charged point charge (blue) and away from the positively charged point charge (red).

[[File:Code_2.png]]

==Examples==

===Simple===
There is an electron at the origin. Calculate the electric field at <4, -3, 1> m.

<table border>
<tr>
<td> <b>Step 1.</b> Find <math>\hat r</math>
Find <math>\vec r_{obs} - \vec r_{electron}:

((4,-3,1) - (0,0,0) = <4,-3,1> </math>m.

Calculate the magnitude of <math>\vec r</math>:

(<math>\sqrt{4^2+(-3)^2+1^2}=\sqrt{26}</math> From <math>\vec r</math>, find the unit vector <math>\hat{r}.</math> <math> <\frac{4}{\sqrt{26}},\frac{-3}{\sqrt{26}},\frac{1}{\sqrt{26}}> </math>
</td></tr>

<tr><td>'''Step 2:''' Find the magnitude of the Electric Field

<math> E_{mag}= \frac{1}{4 \pi \epsilon_0 } \frac{q}{r^2} = \frac{1}{4 \pi \epsilon_0 } \frac{-1.6 * 10^{-19}}{26} </math> </td>
<td> </td></tr>

<td>'''Step 3:''' Multiply the magnitude of the Electric Field by <math>\hat{r}</math> to find the Electric Field
<math>E = \frac{1}{4 \pi \epsilon_0 } \frac{-1.6 * 10^{-19}}{26}*<\frac{4}{\sqrt{26}},\frac{-3}{\sqrt{26}},\frac{1}{\sqrt{26}}>

= <-4.34*10^{-11},3.26*10^{-11},-1.09*10^{-11}> N/C </math></td>
</table border>

===Middling===
A particle of unknown charge is located at <-0.21, 0.02, 0.11> m. Its electric field at point <-0.02, 0.31, 0.28> m is <math><0.124, 0.188, 0.109> </math> N/C. Find the magnitude and sign of the particle's charge.

Given both an observation location and a source location, one can find both r and <math>\hat{r}</math> Given the value of the electric field, one can also find the magnitude of the electric field. Then, using the equation for the magnitude of electric field of a point charge,<math> E_{mag}= \frac{1}{4 \pi \epsilon_0 } \frac{q}{r^2} </math> one can find the magnitude and sign of the charge.

<table border>
<tr>
<td> <b>Step 1.</b> Find <math>\vec r_{obs} - \vec r_{particle} </math>:

<math>\vec r = <-0.02, 0.31, 0.28> m - <-0.21, 0.02, 0.11> m = <0.19,0.29,0.17> m </math>

To find <math>\vec r_{mag} </math>, find the magnitude of <math><0.19,0.29,0.17></math>

<math>\sqrt{0.19^2+0.29^2+0.17^2}=\sqrt{0.1491}= 0.39</math>
</td></tr>

<tr><td><b>Step 2:</b> Find the magnitude of the Electric Field:

<math>E= <0.124, 0.188, 0.109> N/C</math>

<math>E_{mag} = (\sqrt{0.124^2+0.188^2+0.109^2}=\sqrt{0.0626}=0.25</math> </td></tr>

<td>'''Step 3:''' Find '''''q''''' by rearranging the equation for <math>E_{mag}</math>

<math> E_{mag}= \frac{1}{4 \pi \epsilon_0 } \frac{q}{r^2} </math>

By rearranging this equation we get

<math> q= {4 pi * ε_0 } *{r^2}*E_{mag} </math>

<math> q= {1/(9*10^9)} *{0.39^2}*0.25 </math>

<math> q= + 4.3*10^{-12} C </math></td>

</table border>

===Difficult===
The electric force on a -2mC particle at a location (3.98 , 3.98 , 3.98) m due to a particle at the origin is <math>< -5.5*10^{3} , -5.5*10^{3}, -5.5*10^{3}></math> N. What is the charge on the particle at the origin?

Given the force and charge on the particle, one can calculate the surrounding electric field. With this variable found, this problem becomes much like the last one.
<math> E_{mag}= \frac{1}{4 \pi \epsilon_0 } \frac{q}{r_{mag}^2} </math> to find the rmag value. To find <math>\hat r</math> we can find the direction of the electric field as that is obviously going to be in the same direction as <math>\hat r</math>. Then, once we find <math>\hat r</math>, all that is left to do is multiply <math>\hat r</math> by rmag and that will give us the <math> r</math> vector. We can then find the location of the particle as we know <math>r=r_{observation}-r_{particle}</math>

<table border>
<tr>
<td> <b>Step 1.</b> Find the magnitude of the Electric field:
<math> F = Eq </math>
<math> <5.5e3, -7.6e3, 0> = E * -2mC </math>
<math> E = \frac{< -5.5e3 , -5.5e3, -5.5e3>}{-2mC}

= <2.75e6 , 2.75e6, 2.75e6> </math> N/C
</td></tr>

<tr><td><b>Step 2:</b> Find <math>\vec r_{obs} - \vec r_{particle} </math>.

<math>\vec r = <3.98 , 3.98 , 3.98> m - <0 , 0 , 0> m = <3.98 , 3.98 , 3.98> m </math>

To find <math>\vec r_{mag} </math>, find the magnitude of <math><3.98 , 3.98 , 3.98></math>

<math>\sqrt{3.98^2+3.98^2+3.98^2}=\sqrt{47.52}= 6.9</math>
</td></tr>

<td> '''Step 4:''' Find '''''q''''' by rearranging the equation for <math>E_{mag}</math>

<math> E_{mag}= \frac{1}{4 \pi \epsilon_0 } \frac{q}{r^2} </math>

By rearranging this equation we get

<math> q= {4 pi * ε_0 } *{r^2}*E_{mag} </math>

<math> q= {1/(9e9)} *{6.9^{2}}*4.76e6} </math>

<math> q= + 0.253 C </math></td>

</table border>

==Connectedness==
''1. How is this topic connected to something that you are interested in?''

I think the topic is interesting because electric fields are inside the human body and we self-create these fields constantly. It's really cool to think about how your body might be responding and creating these fields continuously and what happens when the balances are thrown off by something (i.e. illness, injury).

''2. How is it connected to your major?''

I'm a business major, but I'm pre-med so it's interesting to think about point charges/electric fields in the macroscopic lens and how they can combine with other forces (magnetic --> electromagnetic forces) to affect your internal systems and processes that occur without you consciously thinking about them.

''3. Is there an interesting industrial application?''

PEMF stands for pulsed electricmagnetic field, which is used by athletes during recovery or physical therapy. There is a difference in cell voltage between healthy cells and diseased cells. While healthy cells maintain a voltage around 70-100 mV, cells with illness have diminished voltage around 40mV. PEMF restores that optimal voltage in damaged cells. Through the utilization of low frequency pulsed electromagnetic fields at a high intensity, voltage in the damaged cells is increased. This stimulates cellular repair and recharges the body’s cells to optimize their performance.

PEMF therapy creates a magnetic field, which increases the movement of ions and electrolytes in the tissues and fluids of the body. The magnetic field helps cells increase ATP production, which restores and / or maintains normal cellular function, speeding up the tissues healing process, repairing damaged tissue, improving circulation, and increasing cellular energy.

==History==
[[File:CoulombCharles300px.jpg]]
''Charles de Coulomb''

Charles de Coulomb was born in June 14, 1736 in central France. He spent much of his early life in the military and was placed in regions throughout the world. He only began to do scientific experiments out of curiously on his military expeditions. However, when controversy arrived with him and the French bureaucracy coupled with the French Revolution, Coulomb had to leave France and thus really began his scientific career.

Between 1785 and 1791, de Coulomb wrote several key papers centered around multiple relations of electricity and magnetism. This helped him develop the principle known as Coulomb's Law, which confirmed that the force between two electrical charges is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. This is the same relationship that is seen in the electric field equation of a point charge.

== See also ==

[[Electric Field]] <br>
[[Electric Force]] <br>
[[Superposition Principle]] <br>
[[Electric Dipole]]

===Further reading===

Principles of Electrodynamics by Melvin Schwartz
ISBN: 9780486134673

Electricity and Magnetism: Edition 3 , Edward M. Purcell David J. Morin

===External links===

Some more information:

*http://hyperphysics.phy-astr.gsu.edu/hbase/electric/epoint.html
*http://www.physics.umd.edu/courses/Phys260/agashe/S10/notes/lecture18.pdf
*https://www.reliantphysicaltherapy.com/services/pulsed-electromagnetic-field-pemf

==References==

Chabay. (2000-2018). ''Matter & Interactions'' (4th ed.). John Wiley & Sons.

PY106 Notes. (n.d.). Retrieved November 27, 2016, from http://physics.bu.edu/~duffy/py106.html

Retrieved November 28, 2016, from http://www.biography.com/people/charles-de-coulomb-9259075#controversy-and-absolution

[[Category:Fields]]

Vectors

2020-10-05T01:57:28Z

Amummareddy3: /* Claimed by: Zach Sanchez (Spring 2020) */

== Claimed by: Ashita Mummareddy (Fall 2020) ==

This page defines and describes vectors.

==The Main Idea==

In mathematics and physics, a vector is a quantity with both a magnitude and a direction in space. The magnitude of a vector is a scalar value which can represent a variety of characteristics, depending on the situation. The magnitude of the vector (and the vector itself) has units corresponding to the characteristic it represents. The direction refers to the angle from the positive x -axis. Thus, the magnitude refers to the length and the direction refers to the angle. Note that a vector does not necessarily extend from one point in real, physical space to another; unless the magnitude is in units of length, the magnitude of the vector usually represents a property that exists at a single point in real, physical space, or at no position in particular.

===A Mathematical Model===

====Simple Examples of Vector Quantities====

To understand what it means for a vector to have both a magnitude and a direction, and to understand why the magnitude and direction together are often considered a single quantity, it can be helpful to consider an example. The [[Velocity]] of an object is an example of a vector quantity. The speed of the object, often given in meters per second (m/s), is a scalar value describing how quickly the object is moving. Speed is the magnitude of the velocity vector. However, the object's movement happens in a particular spatial direction, which the speed alone does not tell us. The direction of the object's movement is also a part of the velocity vector. Together, speed and direction comprise the velocity vector and give a complete description of an object's motion at a point in time. Another example of a vector is position; the distance of a point from the origin of a coordinate system can be represented as the magnitude of a vector, and this distance together with a direction describes exactly where a point can be found.

====Beginning Concepts====

A vector is typically written as a letter with an arrow over it. For example: <math>\vec{a} = \langle 1, 2, 3 \rangle</math>. The arrow may one-sided to make it easier to write. Alternatively, a vector might simply be written as a boldface letter: <b>a</b>. Which letter is used depends on the context; for example, <math>\vec{v}</math> represents velocity.

A specific component of a vector (see the section titled "forms") is denoted by a subscript: c<sub>x</sub>. For example, q<sub>y</sub> represents the y component of some vector <math>\vec{q}</math>.

Surrounding a vector by | symbols denotes its magnitude: <math>|\vec{b}|</math>. Alternatively, there are exists the notation <math>\overline{v}</math> and <math>\lVert \vec{v} \rVert_2</math>, with the latter denoting the Euclidean norm, which is the type of vector norm that you will see most often.

The magnitude of some vector is defined to be <math>\lVert \vec{v} \rVert_2 = \sqrt{v_x^2 + v_y^2 + v_z^2}</math>, where the axes of the 3D-map are mathematically notated as <math>\hat{x}, \hat{y}, \hat{z}</math>.

A unit vector is a vector whose magnitude is equal to one, and normalization is the process of setting a vector's magnitude to 1. A unit vector is denoted by a letter with a ^ symbol (called a "hat") written over it: <math>\hat{d}</math> (read as "d-hat"). Certain letters represent specific unit vectors. For example, <math>\hat{i}</math>, <math>\hat{j}</math>, and <math>\hat{k}</math> (alternatively, <math>\hat{x}</math>, <math>\hat{y}</math>, and <math>\hat{z}</math>) are unit vectors pointing in the +x, +y, and +z directions respectively. Non-Cartesian coordinate systems often have their own unit vectors; for example, 2D polar coordinates make use of the <math>\hat{r}</math> and <math>\hat{\theta}</math> unit vectors.

Unit vectors are defined to be for an arbitrary vector: <math>\hat{v} = \frac{\vec{v}}{\lVert \vec{v} \rVert_2}</math>

====Visually Representing Vectors====

Vectors are visually represented by arrows. The length of the arrow represents the magnitude of the vector, while the direction the arrow points in represents the direction of the vector. If a vector exists at a particular point in space, the "tail" of the arrow (the end without the V shape) should be placed at that point.

This example shows a visual representation of the velocity vector of a ball, which is moving to the right at a speed of 5m/s.

[[File:Vectorvisualrepresentation.png]]

====Important Vector Operations====

It is possible to perform a variety of mathematical operations on vectors, both with other vectors and with scalars. These operations appear in a variety of formulas in physics. To make the operations easier to learn, they are defined below assuming all vectors to be 3-dimensional; the more general n-dimensional definitions look more confusing. If necessary, it is easy to guess how to perform each operation with n-dimensional vectors by extrapolating from the 3-dimensional case.

<u>Addition:</u>

<math>\vec{a} + \vec{b} = \langle (a_x + b_x), (a_y + b_y), (a_z + b_z) \rangle</math>.

In other words, to add two vectors, simply add their like components to form the new components. Visually, if the tail of one vector is placed at the tip of another, their sum will extend from the tail of the second vector to the tip of the first:

[[File:Vectoraddition.png|600px]]

<u>Subtraction:</u>

<math>\vec{a} - \vec{b} = \langle (a_x - b_x), (a_y - b_y), (a_z - b_z)</math>.

In other words, to subtract two vectors, simply subtract their like components to form the new components.

<u>Multiplication by scalar:</u>

<math>k \vec{a} = \langle k \cdot a_x, k \cdot a_y, k \cdot a_z \rangle</math>.

In other words, multiplying a vector by a scalar multiplies each of that vector's components by that scalar. Note that this only affects a vector's magnitude, not its direction, unless the scalar is negative, in which case the direction of the vector is reversed.

[[File:Vectorscalarmultiplication.png|400px]]

<u>Division by scalar:</u>

Division by a scalar behaves exactly like scalar multiplication.

<math>\frac{\vec{a}}{k} = \langle \frac{a_x}{k}, \frac{a_y}{k}, \frac{a_z}{k} \rangle</math>.

<u>Dot product (also called scalar product):</u>

<math>\vec{a}\cdot\vec{b} =(a_x \cdot b_x) + (a_y \cdot b_y) + (a_z \cdot b_z)</math>.

In other words, the dot product of two vectors is the sum of the products of their like components. Note that this is a scalar value.

It is important to note that the dot product of two vectors has a specific value: <math>\vec{a}\cdot\vec{b} = |\vec{a}| \cdot |\vec{b}| \cdot \cos(\theta)</math>, where <math>\theta</math> is the angle between the vectors.

<u>Cross product (also called vector product):</u>

The simplest definition, which is also the one found on the formula sheet, is:

<math>\vec{a} \times \vec{b} = = \langle (a_yb_z - a_zb_y), (a_zb_x - a_xb_z), (a_xb_y - a_yb_x) \rangle = (a_yb_z - a_zb_y)\hat{i} + (a_zb_x - a_xb_z)\hat{j} + (a_xb_y - a_yb_x)\hat{k}</math>.

This is equivalent to the following matrix determinant, which may be easier to remember:

<math>\begin{vmatrix}
\hat{i} & \hat{j} & \hat{k} \\
a_x & a_y & a_z \\
b_x & b_y & b_z
\end{vmatrix}</math>

Note that this is a vector quantity. It is important to note that the <i>magnitude</i> of the cross product of two vectors has a specific value: <math>|\vec{a}\times\vec{b}| = |\vec{a}| \cdot |\vec{b}| \cdot \sin(\theta)</math>, where <math>\theta</math> is the angle between the vectors.

The direction of the cross product of two vectors is perpendicular to the plane in which those vectors lie and is given by the [[Right Hand Rule]]. 2D vectors do not have cross products. While the other operations listed here are commutative, associative (where the associative property is defined), and distributive over addition; cross product multiplication is not associative and is anticommutative (<math>\vec{a}\times\vec{b} = -\vec{b}\times\vec{a}</math>), meaning that if the order of the factors is reversed, their cross product will be reversed in direction.

<b>For more mathematically-advanced students</b>, I will invoke some higher-level linear algebra. Recall that the null space, defined as <math>Null(A) = \{ \vec{v} \in \mathbb{R}^n \mid A\vec{v} = \vec{0} \}</math> for an arbitrary matrix <math>A \in \mathbb{R}^{n \times n}</math>, is orthogonal, or normal, to its row space. That is, for all <math>\vec{v} \in Null(A)</math> and for all columns <math>a_{j:}</math>, we have <math>\sum^n_{i=1} a_{i,j} v_i = 0</math>.

For two vectors <math>\vec{a},\vec{b} \in \mathbb{R}^{3 \times 1}</math> (<i>note that this is only valid for 3x1 vectors</i>), if we think of <math>\vec{a}</math> as corresponding to a row in a matrix and <math>\vec{b}</math> to another, then the null space is equivalent to their cross-product. <b>This is why the right-hand rule works.</b> If we visualize the plane spanned by two vectors with our index and middle fingers, then their null space will be perpendicular to the intersection point of the two vectors, or rather the thumb. Likewise, if we use all of our fingers and our thumb to make an L-shape representing the span of the two vectors, upon curling our fingers, we will have the direction of the cross product, which is orthogonal to the two vectors represented by your thumb and fingers.

====Forms====

The information necessary to describe a specific vector can be presented in several forms.

<u>Magnitude and direction form</u>

In this form, the magnitude and the direction of the vector are explicitly stated. The statement describing direction might be a cardinal direction (ex. "north"), a direction on a graph (ex. "the +x direction"), or an angle (ex. "210<math>^\circ</math> from the x axis counterclockwise"), depending on the situation. Magnitude and direction form is often used in word problems because it is easy for humans to understand.

<u>Component form</u>

In this form, the vector is divided into components, each representing a different coordinate direction. In 2D space, these are the x and y directions. In 3D space, these are the x, y, and z directions. Each component tells how much the vector extends in that particular direction. Often, the three components are written enclosed by angle brackets and separated by commas. For example, the vector <2,0,-3> describes a vector that extends 2 units in the +x direction, 0 units in the y direction, and 3 units in the -z direction. Most vector operations described below can only be performed if the vectors are in component form, so this form may be necessary to do math for certain problems. Furthermore, programming languages store vectors in component form.

<u>Unit vector form</u>

In this form, the vector is expressed as a sum of unit vectors, each corresponding to a different coordinate direction. The symbols <math>\hat{i}</math>, <math>\hat{j}</math>, and <math>\hat{k}</math> OR the symbols <math>\hat{x}</math>, <math>\hat{y}</math>, and <math>\hat{z}</math> are used to represent unit vectors in the x, y, and z directions respectively. Consider the vector <2,0,-3>. It can be expressed in unit vector form as <math>2\hat{i} - 3\hat{k}</math>, meaning 2 times the x direction unit vector minus 3 times the z direction unit vector (see vector operations). While often considered its own form, unit vector form is very similar to component form, as the information describing the vector is stored in the same values. All references to component form in the rest of this page also apply to unit vector form.

[[File:Vectorsdifferentforms.png]]

Note that regardless of which form is used, an n-dimensional vector requires n values to mathematically describe. For example, consider a 3-dimensional vector. Describing this vector in component form requires an x value, a y value, and a z value. Describing it in magnitude and direction form requires one value to give the magnitude of the vector and two to give the its direction (the direction of a 3D vector could be described using, say, its polar angle <math>\theta</math> and its azimuthal angle <math>\phi</math>). A 1-dimensional vector (such as the velocity of a particle whose movement is constrained to the x axis) can be described using only 1 value whose sign indicates the vector's direction.

<u>Converting between forms</u>

It is possible to convert vectors from one form to another using simple trigonometry.

To find the magnitude of a vector in component form, use the Pythagorean theorem: add the squares of the components and take the square root of the result. For a 2D vector, <math>|\vec{a}| = \sqrt{a_x^2 + a_y^2}</math>.

To find the direction of a vector in component form, use inverse trigonometric functions. For a 2D vector, <math>\theta = \tan^{-1}(\frac{a_y}{a_x})</math>, where <math>\theta</math> is the angle vector <math>\vec{a}</math> makes with the x axis in the counterclockwise direction.

To find the components of a vector in magnitude and direction form, use trigonometric functions. For a 2D vector, <math>a_x = |\vec{a}|\cos\theta</math> and <math>a_y = |\vec{a}|\sin\theta</math>, where <math>\theta</math> is the angle vector <math>\vec{a}</math> makes with the x axis in the counterclockwise direction.

===A Computational Model===

In VPython, vector objects are in component form; each one has an x, y, and z component. Recall that in VPython, using the default camera orientation, the +x axis points to the right, the +y axis points upwards, and the +z axis points out of the plane of the screen towards the viewer. The constructor for a vector object is the word "vec" or "vector," and it takes three arguments, which define its x, y and z components respectively. A line to create a vector called "velocity" might look like this:

velocity = vec(3,-1,2)

To access or modify a specific component of a vector object, its name should be followed by a period and an x, y, or z. For example, to change the x component of the above velocity vector from 3 to 5, the following line might be used:

velocity.x = 5

In VPython, vectors have many uses. The position of each object is defined as a vector; the position vector's tail lies at the origin and its head lies at the center of the object in question. Furthermore, the dimensions of a rectangular prism (a "box" object) are defined as a vector; the x component determines its width, the y component its height, and the z component its thickness.

To graphically represent a vector such as an electric field, "arrow" objects should be used. In addition to taking a position vector (which determines the position of the arrow's tail), arrow objects take an "axis" vector, which determines their size and shape. To represent a vector, simply make it the axis of an arrow object.

==Examples==

===Simple===

Vector <math>\vec{a}</math> is <2,4,2>. Vector <math>\vec{b}</math> is <-1,1,3>. What is the magnitude of the vector <math>\vec{a} - 2\vec{b}</math>?

Solution:

<math>\vec{a} - 2\vec{b} = <2,4,2> - 2 * <-1,1,3> </math>

<math> = <2,4,2> - <-2,2,6> </math>

<math> = <4,2,-4></math>

We are asked to find the magnitude of this vector, so let us use the Pythagorean theorem with its components:

<math>|<4,2,-4>| = \sqrt{4^2 + 2^2 + (-4)^2}</math>

<math> = \sqrt{16 + 4 + 16} </math>

<math> = \sqrt{36} </math>

<math> = 6 </math>

===Intermediate===

An airplane is travelling in still air at 240m/s in the direction 35<math>^\circ</math> south of west. A wind begins to blow; the wind has a speed of 80m/s in the direction 15<math>^\circ</math> east of north. What should be the new velocity of the plane relative to the air around it to maintain its original trajectory? You may give your answer in component form (+x is east, +y is north).

[[File:Vectorsplaneproblem.png]]

Solution:

The vector sum of the new velocity of the plane <math>\vec{v_{p,1}}</math> and the velocity of the wind <math>\vec{v_w}</math> should equal the original velocity of the plane <math>\vec{v_{p,0}}</math> (see [[Relative Velocity]]):

[[File:Vectorsplanesolution.png]]

<math> \vec{v_{p,1}} + \vec{v_w} = \vec{v_{p,0}} </math>

<math> \vec{v_{p,1}} = \vec{v_{p,0}} - \vec{v_w} </math>

Let us convert the given vectors to component form for easier subtraction. The +x and +y directions will be east and north respectively.

<math>\vec{v_{p,0}} = <240\cos(215^\circ), 240\sin(215^\circ)></math>m/s (35<math>^\circ</math> south of west is 215<math>^\circ</math> above the x axis)

<math>\vec{v_{p,0}} = <-196.6, -137.7> </math>m/s

<math>\vec{v_w} = <80\cos(75^\circ), 80\sin(75^\circ)></math>m/s (15<math>^\circ</math> east of north is 75<math>^\circ</math> above the x axis)

<math>\vec{v_w} = <20.7, 77.3> </math>m/s

Now let us subtract:

<math> \vec{v_{p,1}} = \vec{v_{p,0}} - \vec{v_w} </math>

<math> \vec{v_{p,1}} = <-196.6, -137.7> - <20.7, 77.3> </math>m/s

<math> \vec{v_{p,1}} = <-217.3, -214.9> </math>m/s

===Difficult===

What is the angle between the vectors <2,5,-2> and <3,-4,-1>?

Solution:

The dot product between two vectors is equal to the product of their magnitudes times the cosine of the angle between them. Let us use this property to find the angle between the given vectors.

<math> <2,5,-2> \cdot <3,-4,-1> = |<2,5,-2>| * |<3,-4,-1>| * \cos\theta </math>

Rearranging this yields

<math> \theta = \cos^{-1}\frac{<2,5,-2> \cdot <3,-4,-1>}{|<2,5,-2>| * |<3,-4,-1>|} </math>

Let us evaluate the dot product and simplify:

<math> \theta = \cos^{-1}\frac{2(3) + 5(-4) + (-2)(-1)}{\sqrt{2^2 + 5^2 + (-2)^2} * \sqrt{3^2 + (-4)^2 + (-1)^2}} </math>

<math> \theta = \cos^{-1}\frac{-12}{\sqrt{33 * 26}} </math>

<math> \theta = 114^\circ </math>

==Connectedness==

Vectors are used in many fields and levels of physics. Kinematics, for example, studies the relationships between the position vector and its time derivatives (velocity, acceleration). Force is also a vector quantity, as are certain system properties such as linear and angular momentum. Vector fields are commonly used in electricity and magnetism, as well as in fluid dynamics.

==History==

It is unknown who first developed the idea of vectors, but the oldest known reference to vectors is in the work <i>Mechanics</i> by Hero of Alexandria (first century AD), which described their addition. At this point, however, the idea of a vector was little more than a line segment with a specific orientation; they had a length extending from one point in physical space to another but were not used to represent anything else.

In the early 19th century, several mathematicians and physicists (including Caspar Wessel (1745-1818), Jean Robert Argand (1768-1822), Carl Friedrich Gauss (1777-1855)), and William Rowan Hamilton (1805-1865) used 2D vectors to represent complex numbers; one component would represent the real value and another would represent the imaginary value. Hamilton would also become the first to use the word "vector." August Ferdinand Möbius (1790-1868) contributed to vector math in his 1827 book <i>The Barycentric Calculus</i>, in which he developed the convention of labeling vectors with letters and defined the multiplication of a vector by a scalar. Hermann Grassmann (1809-1877) wrote in his 1844 book <i>Ausdehnungslehre</i> (German for "The Calculus of Extension") that vectors could exist in space of any number of dimensions and described much of what would become linear algebra, which makes ample use of vectors.

The modern language and conventions surrounding vectors come largely from notes created by J. Willard Gibbs (1839--1903), a professor at Yale University.

==See also==

===External links===
Mathematical Computations on Vectors: [http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf [http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf]

Computational Work with Vectors: [http://vpython.org/contents/docs/vector.html http://vpython.org/contents/docs/vector.html]

Basics of Vectors: [https://www.physics.uoguelph.ca/tutorials/vectors/vectors.html https://www.physics.uoguelph.ca/tutorials/vectors/vectors.html]

===Further Reading===

Vector Analysis by Josiah Willard Gibbs

Introduction to Matrices and Vectors by Jacob T. Schwartz

==References==
[https://www.mathsisfun.com/algebra/vectors.html https://www.mathsisfun.com/algebra/vectors.html]

[http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/1.-vectors-and-matrices/part-a-vectors-determinants-and-planes/session-1-vectors/MIT18_02SC_notes_0.pdf ]

[http://mathinsight.org/vector_introduction http://mathinsight.org/vector_introduction]

[http://www.math.mcgill.ca/labute/courses/133f03/VectorHistory.html http://www.math.mcgill.ca/labute/courses/133f03/VectorHistory.html]

[[Category: Geometry]]