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Physics: High School18 chapters | 211 lessons

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Lesson Transcript

Instructor:
*David Wood*

David has taught Honors Physics, AP Physics, IB Physics and general science courses. He has a Masters in Education, and a Bachelors in Physics.

After watching this video, you will be able to explain what Gauss' Law is, derive Gauss' Law for a sphere and cylinder, and use it to solve problems. A short quiz will follow.

**Gauss' Law** is a law that describes what an electric field will look like due to a known distribution of electric charge. It was first formulated in the 19th century. Gauss' Law also comprises one of the four Maxwell's Equations that describe the force of electromagnetism.

To be more specific, Gauss' Law can be explained in words as: the total of the electric flux out of a closed surface is equal to the magnitude of the charge enclosed divided by the permittivity of free space. As a mathematical equation, it looks like this:

*Q* is the charge enclosed by a surface, *epsilon-zero* is the permittivity of free space, which is just a constant that is always equal to 8.85 x 10^-12, and *phi* is the electric flux through the surface.

But what is electric flux? The **electric flux**, represented by the Greek letter *phi*, is the electric field, *E*, multiplied by the area, *A*, of a surface perpendicular to the field. *Phi* equals *EA*. It's basically the number of field lines that pass through a surface -- more field lines means a larger flux.

So for example, you could use Gauss' Law to figure out the electric field created by a charged conducting sphere. In that case, you have a charge surrounded by a spherical surface. If you wanted to know the total flux, you would take the electric field strength at the surface of the sphere, and multiply it by the surface area of the sphere.

But not all surfaces are spheres. So the exact equation for Gauss' Law varies depending on the particular surface you're looking at.

This might be easier if we went through a few examples. Let's first go back to the sphere.

Gauss' law tells us that the flux is equal to the charge *Q*, over the permittivity of free space, *epsilon-zero*. But flux is also equal to the electric field *E* multiplied by the area of the surface *A*. So *EA* equals *Q* over *epsilon-zero*.

For a sphere, the surface area is given by 4pi*r*^2, so we can plug that in for *A*. And finally, if we rearrange for the electric field *E*, we find that the electric field *E*s is equal to *Q* over 4pi*epsilon-zero**r*^2. This is an expression for the electric field created by a charged sphere.

Okay, let's try another one. How about a conducting cylinder?

We start with the same basic equation. But this time, the area is different. The surface area of a cylinder is equal to 2pi*rL*, where *r* is the radius of the cross-section of the cylinder and *L* is the length of the cylinder. If we rearrange that for *E* again, this time we get *Q* over 2pi*epsilon-zerorL*. So that's an expression for the electric field created by a charged cylinder.

We could do the same thing for any uniformly shaped object, so Gauss' Law is a very powerful tool.

Okay, let's try using one of these equations in a practice problem. Let's imagine we have a metal sphere, with a total charge 0.1 coulombs. Use Gauss' Law to find the electric field strength at a distance of 0.4 meters from the sphere, assuming the sphere has a radius less than 0.4 meters.

Well, first of all, we should write down what we know. The total charge, *Q*, is 0.1 coulombs. The radius, *r*, is 0.4 meters. And *epsilon-zero* is always equal to 8.85 x 10^-12.

Gauss' Law looks like this. And the surface area of a sphere is 4pi*r*^2. Like we said earlier, this re-arranges to form the final equation below. And then all we have to do is plug numbers in and solve.

Typing all that into your calculator, you should come up with an electric field strength of 5.6 x 10^9 newtons per coulomb. And that's it! That's our answer.

**Gauss' Law** is a law that describes what an electric field will look like due to a known distribution of electric charge. To be more specific, Gauss' Law can be explained in words like this: the total of the electric flux out of a closed surface is equal to the magnitude of the charge enclosed divided by the permittivity of free space. As a mathematical equation, it looks like this:

But what is electric flux? The **electric flux**, represented by the Greek letter *phi*, is the electric field, *E*, multiplied by the area, *A*, of a surface perpendicular to the field. *Phi* equals *EA*. It's basically the number of field lines that pass through a surface -- more field lines means a larger flux.

To use Gauss' Law, you have to plug in an expression for the area, *A*. That might be the area of a sphere, 4pi*r*^2, or the area of a cylinder, 2pi*rL*, or something else. You decide the shape based on the shape of the object that contains the charge. Once you do that, you have a final equation you can use to solve problems.

Complete this lesson on Gauss' Law in order to subsequently:

- Recite Gauss' Law and highlight its components
- Understand why this law is useful for determining the electric flux out of differently shaped surfaces

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Physics: High School18 chapters | 211 lessons

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- Electric Force Fields and the Significance of Arrow Direction & Spacing 5:56
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