File:Wiki resource.jpg

2017-11-26T03:39:20Z

Linhtetkyaw07:

Gauss's Law

2017-11-26T03:38:55Z

Linhtetkyaw07: /* Examples */

'''Claimed by Charu Thomas (SPRING 2017)''' Claimed by Lin Htet Kyaw FALL 2017

Gauss's Law is an equation relating the charges and electric flux. It states that net electric flux outside a surface is equal to the charge inside the surface over the constant permittivity of space (ε0 = 8.854187817...×10−12 F⋅m−1). Coulomb's Law and Gauss's Law are intimately connected because Coulomb's Law relates the charge to electric field. Gauss's Law relates electric flux, a quantity that is equal to the product of the perpendicular component of E-field and Area of closed surface.
It is one of Maxwell's Equations, formulated by Carl Friedrich Gauss, a notorious German mathematician.

==The Main Idea==

We know from the previous introduction of flux that there is a relationship between the electric flux on a closed surface and the charges inside the surface. However, we would like to figure out what that particular factor is.

We start by considering a point charge of +Q enclosed by an imaginary spherical shell.

[[File:gaussv2212.jpg]]

Everywhere on the imaginary shell, the electric field produced is parallel to the normal unit vector. Cos(0) = 1, so the dot product evaluates to the magnitude of the electric field. Recall that the surface area of the imaginary sphere is 4*pi*r^2. With Coulomb's Law and the surface area of a sphere, we get that electric flux is equal to +Q/ε0. This implies that the factor is 1/ε0.

Note that if the point charge was negative, the electric field would still be parallel, just opposite direction. Cos(180) = -1 so the dot product would be -1 times the magnitude of the electric field.

The idea of Gauss's Law is that the electric flux out of a closed surface is equivalent to the charge enclosed, divided by the permittivity.

There is a near identical law to this law, known as Gauss's law for Magnetism. The variation found is that magnetic fields are used instead of electric fields in the calculations. Also, Gauss's Law for Gravity is very similar as well. To state it again, the electric flux passing through a closed surface is the same as the charge enclosed, divided by permittivity of the surface. This implies that the electric flux is proportional to the total charge enclosed. Any closed surface can be have Gauss's Law applied to it. For symmetrically shaped objects, Gauss's Law greatly simplifies calculation of electric field enclosed by surface.

===A Mathematical Model===

A very helpful and clear summary of this Law can be found in the diagram below. As can be seen on the left side of this diagram, change in flux equals electric field multiplied by change in area.

Image Taken from Hyperphysics

[[File:Gaulaw.gif]]

To more clearly state it, the formula for this Law is the electric flux equals the total charge contained by a closed surface, divided by the permittivity (epsilon naught: ε0 = 8.854187817...×10−12 F⋅m−1).

[[File:Adc2dff3156800a39ef0a9df76a7d868.png]]

==Examples==

In order to apply Gauss's Law, it is important to be certain you are working with a closed surface, then set electric flux equal to the internal field divided by the permittivity (epsilon naught: ε0 = 8.854187817...×10−12 F⋅m−1). An example of this Law being applied can be found below.

[[File:Gauss_law3.png]]

Below is a further example of Gauss's Law with explanation.

[[File:Gauss_14.jpg]]

This example shows how to find the electric field in a sphere using Gauss's Law.

[[File:gauss's_sphere.jpg]]

[[File:wiki resource.jpg]]

==Connectedness==
Gauss's Law, as well as the other Maxwell Equations form a basis for electrodynamics. They are the fundamental core of this field of study.
Magnetostatics study is also closely related to Gauss's Law, but in particular Gauss's Law of Magnetism, which is very similar to Gauss's Law relating to electric fields.

As an Industrial and Systems Engineering major, we deal with many statistical distributions. A very important fundamental distribution is the Gaussian distribution or the Normal Distribution. With inference, the Gaussian Distribution comes up in confidence intervals for single statistics. With comparison inference, like finding the pairwise difference between statistics, generally we use other distributions such as the T-Distribution. However, the T-Distribution approximates the Gaussian distribution with degrees of freedom greater than 29.

==History==

[[File:220px-Carl_Friedrich_Gauss_(C._A._Jensen).jpg]]

Carl Friedrich Gauss was a German Mathematician and Physicist who contributed notably to a wide variety of fields regarding mathematical and scientific study. He has been referred to as the "greatest mathematician since antiquity" and the "foremost of mathematicians". He is considered one of the most impactful and influential contributors to the fields of Mathematics and Physics in history.

== See also ==

Gauss's Law is tied in closely with the other of Maxwell's equations that can be found here in the Physics Book.

http://physicsbook.gatech.edu/Gauss%27s_Flux_Theorem

http://physicsbook.gatech.edu/Faraday%27s_Law

http://physicsbook.gatech.edu/Magnetic_Flux

http://physicsbook.gatech.edu/Ampere%27s_Law

===External links===

http://physics.info/law-gauss/

https://en.wikipedia.org/wiki/Gauss%27s_law

https://en.wikipedia.org/wiki/Carl_Friedrich_Gauss

==References==

http://hyperphysics.phy-astr.gsu.edu/hbase/electric/gaulaw.html

https://web.iit.edu/sites/web/files/departments/academic-affairs/academic-resource-center/pdfs/Gauss_Law.pdf

spiff.rit.edu

study.com

[[Category:Which Category did you place this in?]]

Gauss's Law

2017-11-25T19:37:30Z

Linhtetkyaw07:

'''Claimed by Charu Thomas (SPRING 2017)''' Claimed by Lin Htet Kyaw FALL 2017

Gauss's Law is an equation relating the charges and electric flux. It states that net electric flux outside a surface is equal to the charge inside the surface over the constant permittivity of space (ε0 = 8.854187817...×10−12 F⋅m−1). Coulomb's Law and Gauss's Law are intimately connected because Coulomb's Law relates the charge to electric field. Gauss's Law relates electric flux, a quantity that is equal to the product of the perpendicular component of E-field and Area of closed surface.
It is one of Maxwell's Equations, formulated by Carl Friedrich Gauss, a notorious German mathematician.

==The Main Idea==

We know from the previous introduction of flux that there is a relationship between the electric flux on a closed surface and the charges inside the surface. However, we would like to figure out what that particular factor is.

We start by considering a point charge of +Q enclosed by an imaginary spherical shell.

[[File:gaussv2212.jpg]]

Everywhere on the imaginary shell, the electric field produced is parallel to the normal unit vector. Cos(0) = 1, so the dot product evaluates to the magnitude of the electric field. Recall that the surface area of the imaginary sphere is 4*pi*r^2. With Coulomb's Law and the surface area of a sphere, we get that electric flux is equal to +Q/ε0. This implies that the factor is 1/ε0.

Note that if the point charge was negative, the electric field would still be parallel, just opposite direction. Cos(180) = -1 so the dot product would be -1 times the magnitude of the electric field.

The idea of Gauss's Law is that the electric flux out of a closed surface is equivalent to the charge enclosed, divided by the permittivity.

There is a near identical law to this law, known as Gauss's law for Magnetism. The variation found is that magnetic fields are used instead of electric fields in the calculations. Also, Gauss's Law for Gravity is very similar as well. To state it again, the electric flux passing through a closed surface is the same as the charge enclosed, divided by permittivity of the surface. This implies that the electric flux is proportional to the total charge enclosed. Any closed surface can be have Gauss's Law applied to it. For symmetrically shaped objects, Gauss's Law greatly simplifies calculation of electric field enclosed by surface.

===A Mathematical Model===

A very helpful and clear summary of this Law can be found in the diagram below. As can be seen on the left side of this diagram, change in flux equals electric field multiplied by change in area.

Image Taken from Hyperphysics

[[File:Gaulaw.gif]]

To more clearly state it, the formula for this Law is the electric flux equals the total charge contained by a closed surface, divided by the permittivity (epsilon naught: ε0 = 8.854187817...×10−12 F⋅m−1).

[[File:Adc2dff3156800a39ef0a9df76a7d868.png]]

==Examples==

In order to apply Gauss's Law, it is important to be certain you are working with a closed surface, then set electric flux equal to the internal field divided by the permittivity (epsilon naught: ε0 = 8.854187817...×10−12 F⋅m−1). An example of this Law being applied can be found below.

[[File:Gauss_law3.png]]

Below is a further example of Gauss's Law with explanation.

[[File:Gauss_14.jpg]]

This example shows how to find the electric field in a sphere using Gauss's Law.

[[File:gauss's_sphere.jpg]]

==Connectedness==
Gauss's Law, as well as the other Maxwell Equations form a basis for electrodynamics. They are the fundamental core of this field of study.
Magnetostatics study is also closely related to Gauss's Law, but in particular Gauss's Law of Magnetism, which is very similar to Gauss's Law relating to electric fields.

As an Industrial and Systems Engineering major, we deal with many statistical distributions. A very important fundamental distribution is the Gaussian distribution or the Normal Distribution. With inference, the Gaussian Distribution comes up in confidence intervals for single statistics. With comparison inference, like finding the pairwise difference between statistics, generally we use other distributions such as the T-Distribution. However, the T-Distribution approximates the Gaussian distribution with degrees of freedom greater than 29.

==History==

[[File:220px-Carl_Friedrich_Gauss_(C._A._Jensen).jpg]]

Carl Friedrich Gauss was a German Mathematician and Physicist who contributed notably to a wide variety of fields regarding mathematical and scientific study. He has been referred to as the "greatest mathematician since antiquity" and the "foremost of mathematicians". He is considered one of the most impactful and influential contributors to the fields of Mathematics and Physics in history.

== See also ==

Gauss's Law is tied in closely with the other of Maxwell's equations that can be found here in the Physics Book.

http://physicsbook.gatech.edu/Gauss%27s_Flux_Theorem

http://physicsbook.gatech.edu/Faraday%27s_Law

http://physicsbook.gatech.edu/Magnetic_Flux

http://physicsbook.gatech.edu/Ampere%27s_Law

===External links===

http://physics.info/law-gauss/

https://en.wikipedia.org/wiki/Gauss%27s_law

https://en.wikipedia.org/wiki/Carl_Friedrich_Gauss

==References==

http://hyperphysics.phy-astr.gsu.edu/hbase/electric/gaulaw.html

https://web.iit.edu/sites/web/files/departments/academic-affairs/academic-resource-center/pdfs/Gauss_Law.pdf

spiff.rit.edu

study.com

[[Category:Which Category did you place this in?]]

Point Charge

2017-11-25T19:06:22Z

Linhtetkyaw07:

This page is all about the [[Electric Field]] due to a Point Charge.'''CLAIMED BY Richard Granger 4-8-2017'''

==The Main Idea==
Every charge creates its own electric field. The two known elementary charges are the proton and electron, both of which are modeled as point charges. The electric field for point charges points radially outward in the case of the proton, and radially inward in the case of the electron. The field is calculated using Coulomb's Law. Indeed, the electric field is defined as a vector field that associates each point in space with the electric force that an infinitesimal test charge would experience. The source of the electric field in this article is the point charge.

===A Mathematical Model of Electric Field due to Point Charge===

The Electric Field of a Point Charge can be found by using the following formula:

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

Which is derived from [[Gauss's Law]], modelling the point charge as a sphere.

<math>{\epsilon_0} </math> is a constant representing vacuum permittivity, the permittivity of free space, or the electric constant, and is approximately <math>8.854*10^{-12}\frac{C^2}{N m^2}</math>

<math>\frac{1}{4 \pi \epsilon_0 } </math> is known as Coulomb's Constant and is approximately <math>8.987*10^{9}\frac{N m^2}{C^2} </math>

'''''r''''' is the magnitude of the distance between the observation location and the source location
, '''''q''''' is the charge of the particle
and <math>\hat r </math> is the unit vector in the direction of the distance from the source location to the observation point.

The force on a given test charge is governed by <math> F = Eq </math> where '''''E''''' is the electric field and '''''q''''' is the charge of a test charge in Coulombs. This can be represented by Coulomb's Law:

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

By solving for the electric field in <math> F = Eq </math> modeled as Coulomb's Law, one obtains the equation for the electric field of the point charge:

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

Below is a visual representation of electric field lines due to a positive and negative point charge

[[File:Point_charge.GIF]]

===A Computational Model===

Below is a link to some code which can help visualize the Electric Field due to a proton.

https://trinket.io/glowscript/725d552305

Notice how the arrows grow by a factor of <math> \frac{1}{r^{2}} </math> as the observation location gets closer to the proton.

==Examples==

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

'''Step 1:''' 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>

'''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>

'''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>

===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.

'''Step 1:'''
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>

'''Step 2:''' 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>

'''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 * epsilon_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>

===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>

'''Step 1''':
Find the magnitude of the Electric field.

<math> F = Eq </math>

<math> <5.5*10^{3}, -7.6*10^{3}, 0> = E * -2mC </math>

<math> E = \frac{< -5.5*10^{3} , -5.5*10^{3}, -5.5*10^{3}>}{-2mC} = <2.75*10^{6} , 2.75*10^{6} , 2.75*10^{6}> </math> N/C

'''Step 2:''':
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>

'''Step 3:''' Find the magnitude of the Electric Field

<math>E= <2.75*10^{6} , 2.75*10^{6} , 2.75*10^{6}> </math> N/C

<math>E_{mag} = (\sqrt{(2.75*10^{6})^2+2.75*10^{6})^2+2.75*10^{6})^2}=\sqrt{2.27*10^{13}}=4.76*10^{6}</math>

'''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 * epsilon_0 } *{r^2}*E_{mag} </math>

<math> q= {1/(9*10^9)} *{6.9^{2}}*4.76*10^{6} </math>

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

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

This topic is the most very basic aspect of physics relating to electricity and magnetism. As someone who is very interested in physics, this topic is important to understanding how more complex ideas that come in later on chapters work.

2.How is it connected to your major?

As a mechanical engineer, static charges don't relate to what I do all that much, but the vector algebra certainly does.

3.Is there an interesting industrial application?

The electric field of a point particle and the [[Superposition Principle]] are integral in understanding the interactions of many charges in various situations, such as a charged sphere or rod. Understanding the way charges behave is important in working with electricity.

==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]]

===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

==References==

Matter and Interactions Vol. II

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

2017-11-23T03:59:38Z

Linhtetkyaw07:

This page is all about the [[Electric Field]] due to a Point Charge.'''CLAIMED BY Richard Granger 4-8-2017'''
Claimed by Lin Htet Kyaw (11/22/2017) FALL 2017

==The Main Idea==
Every charge creates its own electric field. The two known elementary charges are the proton and electron, both of which are modeled as point charges. The electric field for point charges points radially outward in the case of the proton, and radially inward in the case of the electron. The field is calculated using Coulomb's Law. Indeed, the electric field is defined as a vector field that associates each point in space with the electric force that an infinitesimal test charge would experience. The source of the electric field in this article is the point charge.

===A Mathematical Model of Electric Field due to Point Charge===

The Electric Field of a Point Charge can be found by using the following formula:

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

Which is derived from [[Gauss's Law]], modelling the point charge as a sphere.

<math>{\epsilon_0} </math> is a constant representing vacuum permittivity, the permittivity of free space, or the electric constant, and is approximately <math>8.854*10^{-12}\frac{C^2}{N m^2}</math>

<math>\frac{1}{4 \pi \epsilon_0 } </math> is known as Coulomb's Constant and is approximately <math>8.987*10^{9}\frac{N m^2}{C^2} </math>

'''''r''''' is the magnitude of the distance between the observation location and the source location
, '''''q''''' is the charge of the particle
and <math>\hat r </math> is the unit vector in the direction of the distance from the source location to the observation point.

The force on a given test charge is governed by <math> F = Eq </math> where '''''E''''' is the electric field and '''''q''''' is the charge of a test charge in Coulombs. This can be represented by Coulomb's Law:

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

By solving for the electric field in <math> F = Eq </math> modeled as Coulomb's Law, one obtains the equation for the electric field of the point charge:

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

Below is a visual representation of electric field lines due to a positive and negative point charge

[[File:Point_charge.GIF]]

===A Computational Model===

Below is a link to some code which can help visualize the Electric Field due to a proton.

https://trinket.io/glowscript/725d552305

Notice how the arrows grow by a factor of <math> \frac{1}{r^{2}} </math> as the observation location gets closer to the proton.

==Examples==

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

'''Step 1:''' 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>

'''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>

'''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>

===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.

'''Step 1:'''
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>

'''Step 2:''' 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>

'''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 * epsilon_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>

===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>

'''Step 1''':
Find the magnitude of the Electric field.

<math> F = Eq </math>

<math> <5.5*10^{3}, -7.6*10^{3}, 0> = E * -2mC </math>

<math> E = \frac{< -5.5*10^{3} , -5.5*10^{3}, -5.5*10^{3}>}{-2mC} = <2.75*10^{6} , 2.75*10^{6} , 2.75*10^{6}> </math> N/C

'''Step 2:''':
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>

'''Step 3:''' Find the magnitude of the Electric Field

<math>E= <2.75*10^{6} , 2.75*10^{6} , 2.75*10^{6}> </math> N/C

<math>E_{mag} = (\sqrt{(2.75*10^{6})^2+2.75*10^{6})^2+2.75*10^{6})^2}=\sqrt{2.27*10^{13}}=4.76*10^{6}</math>

'''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 * epsilon_0 } *{r^2}*E_{mag} </math>

<math> q= {1/(9*10^9)} *{6.9^{2}}*4.76*10^{6} </math>

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

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

This topic is the most very basic aspect of physics relating to electricity and magnetism. As someone who is very interested in physics, this topic is important to understanding how more complex ideas that come in later on chapters work.

2.How is it connected to your major?

As a mechanical engineer, static charges don't relate to what I do all that much, but the vector algebra certainly does.

3.Is there an interesting industrial application?

The electric field of a point particle and the [[Superposition Principle]] are integral in understanding the interactions of many charges in various situations, such as a charged sphere or rod. Understanding the way charges behave is important in working with electricity.

==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]]

===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

==References==

Matter and Interactions Vol. II

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]]