2018-11-28T04:31:26Z

Esalisbury3:

Improve by by Gabriel Weese(Spring 2018)

== The Big Picture ==
The superposition principle is based on the idea that in a close system, an object receives a net force equal to the sum of all outside forces acting on it. This is commonly used to calculate the net electric field or magnetic field on an object. More specifically, the Superposition Principle states that the net result of multiple vectors acting on a given place and time is equal to the vector sum of each individual vector. For example, think about two waves colliding. When the waves collide, the height of the combined wave is the sum of the height of the two waves. This principle can be seen throughout nature and theory.
===A Mathematical Model===

The Superposition Principle is derived from the properities of additivity and homogeneity for linear systems which are defined in terms of a scalar value of a by the following equations:

<math>F(x_1 + x_2) = F(x_1) + F(x_2)\quad Additivity</math>

<math>aF(x) = F(ax)\quad Homogeneity</math>
[[File:Superposition.gif|thumb|Example of the Superposition of Electric Fields using charges with changing locations]]

The principle can be applied to any linear system and can be used to find the net result of functions, vectors, vector fields, etc. For the topic of introductory physics, it will mainly apply to vectors and vector fields such as electric forces and fields.

If given a number of vectors passing through a certain point, the resultant vector is given by simply adding all the the vectors at that point. For example, for a number of uniform electric fields passing though a single point, the resulting electric field at that point is given by:

<math>\vec{E} = \vec{E}_{1} + \vec{E}_{2} +...+ \vec{E}_{n} = \sum_{i=1}^n\vec{E}_{i}</math>

This same concept can be applied to electric forces as well as to magnetic fields and forces. This is more useful when dealing with the effect that multiple point charges have on each other is an area of void of other electric fields. By adding together each of the contributing electric field vectors at a specific point you can figure out the net contribution that all of the present charges have on the overall electric field experienced at said location.

You can calculate the electric field of a single point charge by using the following equation:
<math>\vec{E} = \frac{1}{4 \pi \epsilon_0}\sum_{i=1}^n\frac{q_i}{r_i^2}\hat{r_i}</math>

Where:

<math>\frac{1}{4 \pi \epsilon_0}</math> is a constant that equals <math>9e9</math>.

<math>q_i</math> is the charge of the particle you are studying.

<math>r_i</math> is the magnitude of the distance between the particle and the observation location.

<math>\hat{r_i}</math> is the vector pointing from the particle to the observation location.

Conceptually is is also important to note that an object cannot exert a force on itself, so the sum of all of the forces does not include the object itself.

==Examples==

When attempting to solve these problems be sure to show all steps in your solution and include diagrams whenever possible.

===Simple - Two Point Charges and One Dimension===
[[File:SuperpositionExample.png]]

If <math>Q_1</math> equals 1e-4 C and <math>Q_2</math> equals 1e-5C, what is the net electric field at the midpoint of both charges?

<div class="toccolours mw-collapsible mw-collapsed">

'''''Click for Solution'''''
<div class="mw-collapsible-content">
We know that the distance of each charge to the midpoint is <math>1m</math>. Since this value is the magnitude of the distance it will serve as our <math>r_i</math> value for both of the electric field formulas.

Our <math>\hat{r_1}</math> value for <math>E_1</math> will be <math>1</math> and the <math>\hat{r_2}</math> value for <math>E_2</math> will be <math>-1</math>, if you use the midpoint as your observation point in the x-direction.

Now that we have all the necessary values to use the electric field formula, we can plug them in.

<math>\vec{E_1} = \frac{1}{4 \pi \epsilon_0}\frac{q_1}{r_1^2}\hat{r_1} = (9e9)\frac{1e-4C}{1m^2}(1m) = 9e5\frac{N}{C}</math>
<math>\vec{E_2} = \frac{1}{4 \pi \epsilon_0}\frac{q_2}{r_2^2}\hat{r_2} = (9e9)\frac{1e-5C}{1m^2}(-1m) = -9e4\frac{N}{C}</math>

By this point, you have the individual contributions of each charge. To get the net contribution of all the charges, you only need to add them.

<math>\vec{E_1}+\vec{E_2}=9e5\frac{N}{C}-9e4\frac{N}{C}= 8.1e5\frac{N}{C}</math>

This means that the net electric field will point towards the right with a magnitude of <math>8.1e5\frac{N}{C}</math>.
</div>
</div>
=== Medium - Three Point charges and two Dimensions ===
[[File:Superposfile2.JPG]]

----
Using k for the constant term
<math> E = Q1 * rhat/ (r)^2 </math>

<math> F = q3 * E </math>

<math> r = (4d, -3d,0> - (0,3d,)) = (4d,-6d,0) </math>

<math> k * Q3 * Q1 * rhat/ (r)^2 </math>

<math> k* Q^2 * 1/(52d^2) * (2/13 * sqrt(13),-3/13 sqrt(13),0) </math>

=== Medium - Three Point charges and two Dimensions===
[[File:Superpos file.JPG]]

----

What is the magnitude of the net force on the charge -q'?

To begin, we recognize that this problem will be using the superposition principle. It is additionally
important to recognize that objects can't exert forces on themselves. Keeping these in mind, The force exerted on -q' will be a combination of the forces exerted by 3q and negative q.

For this problem, let's consider -q' to be q3, 3q to be q1, and -1 to be q2 and, For the sake of simplicity, we are going to call our constant 'k'

Let's find the forces separately then add them together:

<math> k * (q1*q3)/(r)^2 * rhat </math>

Now that we have the general form of the equation, we plug in variables.

<math> F1 = k * 3q(-q')/(sqrt(3) * l)^2 * (sin(theta),cos(theta),0) </math>
<math>

= k * (-q*q')/ (l^2) * (sin(theta),cos(theta),0) </math>

Great! Now we have the first force from q1, now we need to find the force from q2

<math> F2 = k * (q2*q3)/ (r)^2 * rhat </math>

<math> = k * (q*q)/(l^2) * (-sin(theta), cos(theta),0) </math>

Now that we've found both forces, we add them together.

<math> Fnet = F1+F2 </math>
<math> = k * (-q*q')/ (l^2) * (sin(theta),cos(theta),0) + k * (q*q)/(l^2) * (-sin(theta), cos(theta),0) </math>
<math> = k * (q*q')/(l^2) * (-2sin(theta),0,0) </math>

Once we were able to recognize our problem, it became as simple as plugging in variables and using algebra to solve the problem.

Additionally, if we were asked "What is the direction of the force on the charge -q'?"

Looking back at our work, we can see that our vertical components cancel out, leaving
<math> -2sin(theta) </math>
Indicating our direction is negative x.

We could get to this conclusion without looking at the math by understanding that electric field falls off in a way that resembles
<math> 1/(r)^2 </math>

For q1, it is sqrt(3) times farther than q2, and 3 times the charge. Which is equivalent to
<math> 3*q/(sqrt(3))^2 </math>
Which is equivalent to q2 in the vertical direction. In the horizontal direction, q2 is slightly closer, meaning that the total net charge should be in the negative x direction.

===Medium - Two Point Charges and Two Dimensions===
[[File:BrooksEx1.png]]

If <math>Q_1</math> and <math>Q_2</math> are positive point charges with a charge of e, what is the net electric field at point P?

<div class="toccolours mw-collapsible mw-collapsed">
'''''Click for Solution'''''
<div class="mw-collapsible-content">
To begin this problem, the first step is to find <math>r_1</math> and <math>r_2</math>, the vectors from the charges to point P as well as their magnitudes:

<math>\vec{r_1} = 3\hat{i}+2\hat{j}-(0\hat{i}+0\hat{j})\Rightarrow\vec{r_1} = 3\hat{i}+2\hat{j}</math>

<math>||\vec{r_1}|| = \sqrt{3^2 + 2^2} =\sqrt{13}</math>

<math>\vec{r_2} = 3\hat{i}+2\hat{j}-(4\hat{i}+0\hat{j})\Rightarrow\vec{r_2} = -1\hat{i}+2\hat{j}</math>

<math>||\vec{r_2}|| = \sqrt{-1^2 + 2^2} =\sqrt{5}</math>

Using these in the equation for an electric field from a point charge, you get:

<math>\vec{E_1} = \frac{1}{4 \pi \epsilon_0}\frac{Q_1}{||r_1||^2}\hat{r_1}=\frac{1}{4 \pi \epsilon_0}\frac{e}{13}<\frac{3}{\sqrt{13}}\hat{i}+\frac{2}{\sqrt{13}}\hat{j}> = <9.21E-11\hat{i}+6.14E-11\hat{j}> </math>N/C

<math>\vec{E_2} = \frac{1}{4 \pi \epsilon_0}\frac{Q_2}{||r_2||^2}\hat{r_2}=\frac{1}{4 \pi \epsilon_0}\frac{e}{5}<\frac{-1}{\sqrt{5}}\hat{i}+\frac{2}{\sqrt{5}}\hat{j}> = <-1.29E-10\hat{i}+2.58E-10\hat{j}> </math>N/C

Then, simply add the two electric fields together:

<math>\vec{E}=\vec{E_1}+\vec{E_2} = <-3.69E-11\hat{i}+3.19E-10\hat{j}></math>
</div>
</div>

===Difficult - Five Point Charges and Three Dimensions===

[[File:brooksEx2.png]]

If all point charges have a charge of e, what the the net electric field present at point L?
<div class="toccolours mw-collapsible mw-collapsed">
'''''Click for Solution'''''
<div class="mw-collapsible-content">
This problem is similar to the previous examples but this one includes the z axis and more points. Since there are 5 points and you have already had some practice, you will only see the procedure for the first one. We will still work through the whole problem. Again, first each vector and magnitude:

<math>\vec{d_5} = 0\hat{i}+0\hat{j}+0\hat{k}-(2\hat{i}-1\hat{j}-1\hat{k})-\Rightarrow\vec{d_5} = -2\hat{i}+1\hat{j}+1\hat{k}</math>

<math>||\vec{d_5}|| = \sqrt{(-2)^2 + 1^2 + 1^2} = 2</math>

Do the same for each of the other point charges and plug them into the electric field formula:

<math>\vec{E_5} = \frac{1}{4 \pi \epsilon_0}\frac{Q_5}{||d_5||^2}\hat{d_5}=\frac{1}{4 \pi \epsilon_0}\frac{e}{4}<\frac{-2}{2}\hat{i}+\frac{1}{2}\hat{j}+\frac{1}{2}\hat{k}> = <-3.6E-10\hat{i}+1.8E-10\hat{j}+1.8E-10\hat{k}></math>N/C

Follow the same steps for the other electric fields and add them all together to get your final answer:

<math>\vec{E_1} = <1.44E-9\hat{i}+0\hat{j}+0\hat{k}></math>N/C

<math>\vec{E_2} = <0\hat{i}-1.44E-9\hat{j}+0\hat{k}></math>N/C

<math>\vec{E_3} = <-1.29E-10\hat{i}+0\hat{j}+2.58E-10\hat{k}></math>N/C

<math>\vec{E_4} = <0\hat{i}+0\hat{j}-1.44E-9\hat{k}></math>N/C

<math>\vec{E} = \vec{E_1}+\vec{E_2}+\vec{E_3}+\vec{E_4}+\vec{E_5}=<9.51E-10\hat{i}-1.26E-9\hat{j}+1.00E-9\hat{k}></math>N/C
</div>
</div>

==Connectedness==
[[File:Superposition Example.PNG|thumb|left|A diagram of a circuit you will be able to analyze further on by using the superposition of currents.]]
The superposition principle is critical as you progress to more complicated topics in Physics. This principle simplifies many topics that you will encounter further ahead, by assuming that their net contribution will be equal to the addition of all the individual ones. You will use superposition with electric fields and forces, as well as magnetic fields and forces. In addition, if in a problem you are only given the final net contribution of any type of field, you can inversely solve it by constructing equations that relate the individual formulas. You can then plug in values, and subtract in order to find the magnitudes of each individual contributor. This will also enable you to better analyze different types of circuits.

However, the superposition principle also aids outside of the world of abstract electromagnetism. It is also useful for static problems, which you might encounter if you are a Civil or Mechanical Engineer, or systems and circuitry, which you might encounter if you are a Biomedical engineer.

==History==
[[File:Danielbernoulli.jpg|thumb|right|Daniel Bernoulli]]
The superposition principle was supposedly first stated by [[Daniel Bernoulli]], a famous scientist known for his work in fluid mechanics, statistics, and the Bernoulli Principle. In 1753, he stated that, "The general motion of a vibrating system is given by a superposition of its proper vibrations." This idea was at first rejected by some other popular scientists until it became widely accepted due to the work of Joseph Fourier, a famous scientist known for his work on the Fourier Series, which is used in heat transfer and vibrations, as well as his discovery of the Greenhouse Effect.

== See also ==

===Next Topics===
Now that you have a solid understanding of the Superposition Principle, you can move on to these topics:
*[[Electric Field]]
**[[Point Charge]]
**[[Electric Dipole]]
*[[Electric Force]]
*[[Magnetic Field]]
*[[Magnetic Force]]
*[[Superposition Principle]]
===Further reading===

Books, Articles or other print media on this topic

[http://www.acoustics.salford.ac.uk/feschools/waves/super.php Superposition of Waves]

[http://whatis.techtarget.com/definition/Schrodingers-cat Schrodingers' Cat from a Superposition Point of View]

===External Links===

[https://www.youtube.com/watch?v=p3Xugztl9Ho Visual Explanation of Superposition in Electric Fields]

[https://www.youtube.com/watch?v=mdulzEfQXDE Electric Fields and Superposition Principle Crash Course Video]

[https://www.youtube.com/watch?v=S1TXN1M9t18 Instructional video on how to calculate the net electric field using the superposition principle]

[https://www.youtube.com/watch?v=lCM5dql_ul0 Additional Video on calculating superposition principle]

[https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=1479978&tag=1 Using superposition principle to analyze solar cells]

==References==
[http://physics-help.info/physicsguide/electricity/electric_field.shtml Electric Fields]

[https://en.wikipedia.org/wiki/Superposition_principle Superposition Principle]

[https://en.wikipedia.org/wiki/Daniel_Bernoulli Daniel Bernoulli Biography]

[https://en.wikipedia.org/wiki/Joseph_Fourier Joseph Fourier Biography]
[[Category:Fields]]

File:Gauss's Law1234.jpeg

2018-04-19T02:15:12Z

Esalisbury3:

Gauss's Law

2018-04-19T02:14:43Z

Esalisbury3: /* Examples */

'''Claimed by Charu Thomas (SPRING 2017)''' Claimed by Lin Htet Kyaw FALL 2017; Claimed by Ishita Mathur Spring 2018; Claimed by Eric Salisbury Spring 2018; Claimed by Patricia Estrada Spring 2018

Gauss's Law is found throughout science, technology, engineering, and mathematics. In fact, it is a defining principal in electromagnetism. It was first discovered by Joseph-Louis Lagrange in 1773 and furthered by Carl Friedrich Gauss in 1813. At it's core, it describes the relationship between charges and electric flux. The Law presents that the net electric flux outside a surface is equal to the charge inside the surface over the constant permittivity of space. In equation form it looks like, Φ=Q/εₒ. In addition, this law also relates the integral over Electric field enclosed multiplied by the change in area to net electric flux outside a surface. In equation form this looks like Φ=∮E*dA. This monumental discovery has been the foundation for current physics for over 200 years.

Additionally, Coulomb's Law and Gauss's Law are innately connected. Coulomb's Law relates charge to electric field. Gauss's Law relates charge to electric flux. Both laws, relate charge to an electrical property. For clarification, the electric flux is a quantity that is equal to the product of the perpendicular component of E-field and the area of the closed surface. Not only do these two laws look similar at the surface, but using one law it is quite easy to derive the other law.

==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.

To summarize, 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.

===A Mathematical Model===

To understand this law more fully, one can look at 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. This is the basis for the equation ф=∮E da. When you integrate the change of A, you get the flux.

Image Taken from Hyperphysics

[[File:Gaulaw.gif]]

To more clearly state it, in integral form, 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]]

Additionally, Gauss's law can be written in differential form.

[[File:Screen Shot 2018-04-18 at 5.31.54 PM.png]]

In this equation, ρ is electric charge density, ∇*E is divergence of electric field, and ε0 = 8.854187817...×10−12 F⋅m−1.

The picture below illustrates Gauss's Law with a detailed explanation.

[[File:Gauss_14.jpg]]

==Examples==

'''Easy Example'''

Looking at the images on the right hand side of the screen, we find an easy example in which we will find the electric field in a uniformly charged plate (+Q).

1. Recall Gauss's Formula:

[[File:Gauss for.jpg|thumb|Gauss for]]

2. Now look carefully at the diagram below and go through each of the surfaces one by one. Ask yourself, in what direction does the normal vector point?

[[File:Gauss ex.jpg|thumb|Gauss ex]]

3. Therefore, in the sides where there is 90 degree angle made with E, the flux will just be zero!

[[File:Gauss ex.jpg|thumb|Gauss ex]]

4. In the only case where the flux is not zero is in the sides A and B of the box (which are the same in magnitude). We compute the flux as follows:

[[File:Gauss ans.jpg|thumb|Gauss ans]]

Remember, when doing Gauss's problems, always think about you n vector before diving into the formula.

'''Intermediate Examples:'''

Example 1: The example below shows how to determine the net charge located inside a box and is fundamental to understanding Gauss's Law. It demonstrates the idea that to calculate the net charge inside a box, we have to start off by calculating the fluxes acting on all surfaces on the box by using the Gauss's Law.

[[File:Wiki resource.jpg]]

Example 2: The example below shows how to calculate the net charge enclosed by a box.

[[File:Wiki resource2.jpg]]

Example 3: 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]]

Example 4: In this example, the Qenc is equal to zero. This was determined by adding up all of the fluxes through every surface. This can also be determined by seeing that the same fluxes travel through the same areas of the tent. The equation Ф=∮B da=0 helps explain this when considering that Ф is the same in both equations.

[[File:Gauss's_Law1234.jpeg]]

== Headline text ==

==Connectedness==
Gauss's Law can be found in many areas of science, technology, engineering, and mathematics. In fact, Gauss's Law and other Maxwell Equations form a basis for electrodynamics. They are the fundamental core of this field of study. Gauss's Law is also connected to the Gaussian/Normal distribution, which is a continuous probability distribution that is characterized by a bell-shaped curve. This distribution is found throughout statistics and probability and is used everyday by people in STEM fields.

As Industrial and Systems Engineering majors, 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.

As civil and environmental engineering majors, we also deal with the Gaussian/Normal Distribution in our fields. As an example, the speed data of traffic on a highway is said to follow the normal distribution.

Additional to many engineering majors. Gauss's Law has applications in many different STEM job fields. Engineering majors learn about the different Gaussian laws in college and apply them in there jobs. Millions of people are applying his laws to the world and will continue to do so. Gauss has had a huge impact on the world and will continue to have a big impact on the future. Gauss has done great work with his work.

==History==

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

Carl Friedrich Gauss was a German Mathematician and Physicist. He contributed to a wide variety of fields especially in mathematical and scientific study. In fact his contributions are so great that he is referred to as the "greatest mathematician since antiquity" and the "foremost of mathematicians". These names highlight his illustriousness in those fields as 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/

http://teacher.nsrl.rochester.edu/phy122/Lecture_Notes/Chapter24/Chapter24.html

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

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

http://physicscatalyst.com/elec/guass_0.php

==References==

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

http://www.pas.rochester.edu/~stte/phy114S09/lectures/lect04.pdf

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

http://www.sciencedirect.com/science/article/pii/S2090447911000165

spiff.rit.edu

study.com

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

Gauss's Law

2018-04-19T02:11:40Z

Esalisbury3: /* Examples */

'''Claimed by Charu Thomas (SPRING 2017)''' Claimed by Lin Htet Kyaw FALL 2017; Claimed by Ishita Mathur Spring 2018; Claimed by Eric Salisbury Spring 2018; Claimed by Patricia Estrada Spring 2018

Gauss's Law is found throughout science, technology, engineering, and mathematics. In fact, it is a defining principal in electromagnetism. It was first discovered by Joseph-Louis Lagrange in 1773 and furthered by Carl Friedrich Gauss in 1813. At it's core, it describes the relationship between charges and electric flux. The Law presents that the net electric flux outside a surface is equal to the charge inside the surface over the constant permittivity of space. In equation form it looks like, Φ=Q/εₒ. In addition, this law also relates the integral over Electric field enclosed multiplied by the change in area to net electric flux outside a surface. In equation form this looks like Φ=∮E*dA. This monumental discovery has been the foundation for current physics for over 200 years.

Additionally, Coulomb's Law and Gauss's Law are innately connected. Coulomb's Law relates charge to electric field. Gauss's Law relates charge to electric flux. Both laws, relate charge to an electrical property. For clarification, the electric flux is a quantity that is equal to the product of the perpendicular component of E-field and the area of the closed surface. Not only do these two laws look similar at the surface, but using one law it is quite easy to derive the other law.

==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.

To summarize, 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.

===A Mathematical Model===

To understand this law more fully, one can look at 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. This is the basis for the equation ф=∮E da. When you integrate the change of A, you get the flux.

Image Taken from Hyperphysics

[[File:Gaulaw.gif]]

To more clearly state it, in integral form, 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]]

Additionally, Gauss's law can be written in differential form.

[[File:Screen Shot 2018-04-18 at 5.31.54 PM.png]]

In this equation, ρ is electric charge density, ∇*E is divergence of electric field, and ε0 = 8.854187817...×10−12 F⋅m−1.

The picture below illustrates Gauss's Law with a detailed explanation.

[[File:Gauss_14.jpg]]

==Examples==

'''Easy Example'''

Looking at the images on the right hand side of the screen, we find an easy example in which we will find the electric field in a uniformly charged plate (+Q).

1. Recall Gauss's Formula:

[[File:Gauss for.jpg|thumb|Gauss for]]

2. Now look carefully at the diagram below and go through each of the surfaces one by one. Ask yourself, in what direction does the normal vector point?

[[File:Gauss ex.jpg|thumb|Gauss ex]]

3. Therefore, in the sides where there is 90 degree angle made with E, the flux will just be zero!

[[File:Gauss ex.jpg|thumb|Gauss ex]]

4. In the only case where the flux is not zero is in the sides A and B of the box (which are the same in magnitude). We compute the flux as follows:

[[File:Gauss ans.jpg|thumb|Gauss ans]]

Remember, when doing Gauss's problems, always think about you n vector before diving into the formula.

'''Intermediate Examples:'''

Example 1: The example below shows how to determine the net charge located inside a box and is fundamental to understanding Gauss's Law. It demonstrates the idea that to calculate the net charge inside a box, we have to start off by calculating the fluxes acting on all surfaces on the box by using the Gauss's Law.

[[File:Wiki resource.jpg]]

Example 2: The example below shows how to calculate the net charge enclosed by a box.

[[File:Wiki resource2.jpg]]

Example 3: 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]]

Example 4: In this example, the Qenc is equal to zero. This was determined by adding up all of the fluxes through every surface. This can also be determined by seeing that the same fluxes travel through the same areas of the tent. The equation Ф=∮B da=0 helps explain this when considering that Ф is the same in both equations.

[[Media:Gauss's Law (12343).jpeg]]

== Headline text ==

==Connectedness==
Gauss's Law can be found in many areas of science, technology, engineering, and mathematics. In fact, Gauss's Law and other Maxwell Equations form a basis for electrodynamics. They are the fundamental core of this field of study. Gauss's Law is also connected to the Gaussian/Normal distribution, which is a continuous probability distribution that is characterized by a bell-shaped curve. This distribution is found throughout statistics and probability and is used everyday by people in STEM fields.

As Industrial and Systems Engineering majors, 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.

As civil and environmental engineering majors, we also deal with the Gaussian/Normal Distribution in our fields. As an example, the speed data of traffic on a highway is said to follow the normal distribution.

Additional to many engineering majors. Gauss's Law has applications in many different STEM job fields. Engineering majors learn about the different Gaussian laws in college and apply them in there jobs. Millions of people are applying his laws to the world and will continue to do so. Gauss has had a huge impact on the world and will continue to have a big impact on the future. Gauss has done great work with his work.

==History==

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

Carl Friedrich Gauss was a German Mathematician and Physicist. He contributed to a wide variety of fields especially in mathematical and scientific study. In fact his contributions are so great that he is referred to as the "greatest mathematician since antiquity" and the "foremost of mathematicians". These names highlight his illustriousness in those fields as 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/

http://teacher.nsrl.rochester.edu/phy122/Lecture_Notes/Chapter24/Chapter24.html

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

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

http://physicscatalyst.com/elec/guass_0.php

==References==

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

http://www.pas.rochester.edu/~stte/phy114S09/lectures/lect04.pdf

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

http://www.sciencedirect.com/science/article/pii/S2090447911000165

spiff.rit.edu

study.com

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

Gauss's Law

2018-04-19T02:06:45Z

Esalisbury3: /* Examples */

'''Claimed by Charu Thomas (SPRING 2017)''' Claimed by Lin Htet Kyaw FALL 2017; Claimed by Ishita Mathur Spring 2018; Claimed by Eric Salisbury Spring 2018; Claimed by Patricia Estrada Spring 2018

Gauss's Law is found throughout science, technology, engineering, and mathematics. In fact, it is a defining principal in electromagnetism. It was first discovered by Joseph-Louis Lagrange in 1773 and furthered by Carl Friedrich Gauss in 1813. At it's core, it describes the relationship between charges and electric flux. The Law presents that the net electric flux outside a surface is equal to the charge inside the surface over the constant permittivity of space. In equation form it looks like, Φ=Q/εₒ. In addition, this law also relates the integral over Electric field enclosed multiplied by the change in area to net electric flux outside a surface. In equation form this looks like Φ=∮E*dA. This monumental discovery has been the foundation for current physics for over 200 years.

Additionally, Coulomb's Law and Gauss's Law are innately connected. Coulomb's Law relates charge to electric field. Gauss's Law relates charge to electric flux. Both laws, relate charge to an electrical property. For clarification, the electric flux is a quantity that is equal to the product of the perpendicular component of E-field and the area of the closed surface. Not only do these two laws look similar at the surface, but using one law it is quite easy to derive the other law.

==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.

To summarize, 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.

===A Mathematical Model===

To understand this law more fully, one can look at 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. This is the basis for the equation ф=∮E da. When you integrate the change of A, you get the flux.

Image Taken from Hyperphysics

[[File:Gaulaw.gif]]

To more clearly state it, in integral form, 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]]

Additionally, Gauss's law can be written in differential form.

[[File:Screen Shot 2018-04-18 at 5.31.54 PM.png]]

In this equation, ρ is electric charge density, ∇*E is divergence of electric field, and ε0 = 8.854187817...×10−12 F⋅m−1.

The picture below illustrates Gauss's Law with a detailed explanation.

[[File:Gauss_14.jpg]]

==Examples==

'''Easy Example'''

Looking at the images on the right hand side of the screen, we find an easy example in which we will find the electric field in a uniformly charged plate (+Q).

1. Recall Gauss's Formula:

[[File:Gauss for.jpg|thumb|Gauss for]]

2. Now look carefully at the diagram below and go through each of the surfaces one by one. Ask yourself, in what direction does the normal vector point?

[[File:Gauss ex.jpg|thumb|Gauss ex]]

3. Therefore, in the sides where there is 90 degree angle made with E, the flux will just be zero!

[[File:Gauss ex.jpg|thumb|Gauss ex]]

4. In the only case where the flux is not zero is in the sides A and B of the box (which are the same in magnitude). We compute the flux as follows:

[[File:Gauss ans.jpg|thumb|Gauss ans]]

Remember, when doing Gauss's problems, always think about you n vector before diving into the formula.

'''Intermediate Examples:'''

Example 1: The example below shows how to determine the net charge located inside a box and is fundamental to understanding Gauss's Law. It demonstrates the idea that to calculate the net charge inside a box, we have to start off by calculating the fluxes acting on all surfaces on the box by using the Gauss's Law.

[[File:Wiki resource.jpg]]

Example 2: The example below shows how to calculate the net charge enclosed by a box.

[[File:Wiki resource2.jpg]]

Example 3: 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]]

Example 4: In this example, the Qenc is equal to zero. This was determined by adding up all of the fluxes through every surface. This can also be determined by seeing that the same fluxes travel through the same areas of the tent. The equation Ф=∮B da=0 helps explain this when considering that Ф is the same in both equations.

[[File:Gauss's Law (12343).jpeg]]

== Headline text ==

==Connectedness==
Gauss's Law can be found in many areas of science, technology, engineering, and mathematics. In fact, Gauss's Law and other Maxwell Equations form a basis for electrodynamics. They are the fundamental core of this field of study. Gauss's Law is also connected to the Gaussian/Normal distribution, which is a continuous probability distribution that is characterized by a bell-shaped curve. This distribution is found throughout statistics and probability and is used everyday by people in STEM fields.

As Industrial and Systems Engineering majors, 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.

As civil and environmental engineering majors, we also deal with the Gaussian/Normal Distribution in our fields. As an example, the speed data of traffic on a highway is said to follow the normal distribution.

Additional to many engineering majors. Gauss's Law has applications in many different STEM job fields. Engineering majors learn about the different Gaussian laws in college and apply them in there jobs. Millions of people are applying his laws to the world and will continue to do so. Gauss has had a huge impact on the world and will continue to have a big impact on the future. Gauss has done great work with his work.

==History==

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

Carl Friedrich Gauss was a German Mathematician and Physicist. He contributed to a wide variety of fields especially in mathematical and scientific study. In fact his contributions are so great that he is referred to as the "greatest mathematician since antiquity" and the "foremost of mathematicians". These names highlight his illustriousness in those fields as 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/

http://teacher.nsrl.rochester.edu/phy122/Lecture_Notes/Chapter24/Chapter24.html

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

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

http://physicscatalyst.com/elec/guass_0.php

==References==

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

http://www.pas.rochester.edu/~stte/phy114S09/lectures/lect04.pdf

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

http://www.sciencedirect.com/science/article/pii/S2090447911000165

spiff.rit.edu

study.com

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

File:IMG 0451.jpeg

2018-04-19T02:05:36Z

Esalisbury3:

2018-04-19T01:35:07Z

Esalisbury3: /* Connectedness */

'''Claimed by Charu Thomas (SPRING 2017)''' Claimed by Lin Htet Kyaw FALL 2017; Claimed by Ishita Mathur Spring 2018; Claimed by Eric Salisbury Spring 2018; Claimed by Patricia Estrada Spring 2018

Gauss's Law is physical principal in electromagnetism. It was first discovered by Joseph-Louis Lagrange in 1773 and furthered by Carl Friedrich Gauss in 1813. At it's core, it describes the relationship between charges and electric flux. The Law presents that the net electric flux outside a surface is equal to the charge inside the surface over the constant permittivity of space. In equation form it looks like, Φ=Q/εₒ. In addition, this law also relates the integral over Electric field enclosed multiplied by the change in area to net electric flux outside a surface. In equation form this looks like Φ=∮E*dA. This monumental discovery has been the foundation for current physics for over 200 years.

Additionally, Coulomb's Law and Gauss's Law are innately connected. Coulomb's Law relates charge to electric field. Gauss's Law relates charge to electric flux. Both laws, relate charge to an electrical property. For clarification, the electric flux is a quantity that is equal to the product of the perpendicular component of E-field and the area of the closed surface. Not only do these two laws look similar at the surface, but using one law it is quite easy to derive the other law.

==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.

To summarize, 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.

===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, in integral form, 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]]

Additionally, Gauss's law can be written in differential form.

[[File:Screen Shot 2018-04-18 at 5.31.54 PM.png]]

In this equation, ρ is electric charge density, ∇*E is divergence of electric field, and ε0 = 8.854187817...×10−12 F⋅m−1.

The picture below illustrates Gauss's Law with a detailed explanation.

[[File:Gauss_14.jpg]]

==Examples==

'''Easy Example'''

Looking at the images on the right hand side of the screen, we find an easy example in which we will find the electric field in a uniformly charged plate (+Q).

1. Recall Gauss's Formula:

[[File:Gauss for.jpg|thumb|Gauss for]]

2. Now look carefully at the diagram below and go through each of the surfaces one by one. Ask yourself, in what direction does the normal vector point?

[[File:Gauss ex.jpg|thumb|Gauss ex]]

3. Therefore, in the sides where there is 90 degree angle made with E, the flux will just be zero!

[[File:Gauss ex.jpg|thumb|Gauss ex]]

4. In the only case where the flux is not zero is in the sides A and B of the box (which are the same in magnitude). We compute the flux as follows:

[[File:Gauss ans.jpg|thumb|Gauss ans]]

Remember, when doing Gauss's problems, always think about you n vector before diving into the formula.

'''Intermediate Examples:'''

Example 1: The example below shows how to determine the net charge located inside a box and is fundamental to understanding Gauss's Law. It demonstrates the idea that to calculate the net charge inside a box, we have to start off by calculating the fluxes acting on all surfaces on the box by using the Gauss's Law.

[[File:Wiki resource.jpg]]

Example 2: The example below shows how to calculate the net charge enclosed by a box.

[[File:Wiki resource2.jpg]]

Example 3: 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]]

Example 4:

[[File:IMG_0451.jpeg]]

In this example, the Qenc is equal to zero. This was determined by adding up all of the fluxes through every surface. This can also be determined by seeing that the same fluxes travel through the same areas of the tent.

== Headline text ==

==Connectedness==
Gauss's Law can be found in many areas of science, technology, engineering, and mathematics. In fact, Gauss's Law and other Maxwell Equations form a basis for electrodynamics. They are the fundamental core of this field of study. Gauss's Law is also connected to the Gaussian/Normal distribution, which is a continuous probability distribution that is characterized by a bell-shaped curve. This distribution is found throughout statistics and probability and is used everyday by people in STEM fields.

As Industrial and Systems Engineering majors, 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.

As civil and environmental engineering majors, we also deal with the Gaussian/Normal Distribution in our fields. As an example, the speed data of traffic on a highway is said to follow the normal distribution.

==History==

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

Carl Friedrich Gauss was a German Mathematician and Physicist. He contributed to a wide variety of fields especially in mathematical and scientific study. In fact his contributions are so great that he is referred to as the "greatest mathematician since antiquity" and the "foremost of mathematicians". These names highlight his illustriousness in those fields as 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/

http://teacher.nsrl.rochester.edu/phy122/Lecture_Notes/Chapter24/Chapter24.html

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

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

http://physicscatalyst.com/elec/guass_0.php

==References==

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

http://www.pas.rochester.edu/~stte/phy114S09/lectures/lect04.pdf

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

http://www.sciencedirect.com/science/article/pii/S2090447911000165

spiff.rit.edu

study.com

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