# Gauss's Law applied to an infinite sheet of charge

• ecgprc
In summary, the problem is trying to find the electric field due to an infinite sheet of charge with uniform charge density σ. The character answer is σ/2ε (epsilon nought), but the solution is not clear. To find the field, you examine a small area of the sheet and a small amount of charge enclosed within that area, and find the field produced. The electric field is constant for the sheet, and the results of this calculation apply to the whole sheet.
ecgprc

## Homework Statement

The problem that inspired the upcoming question is: Find the magnitude of the electric field due to an infinite sheet of charge with uniform charge density σ using gauss's law.

I have in fact arrived at the character answer σ/2ε (epsilon nought), but I don't understand this solution. To elaborate, I'm unsure as to why I was able to solve this problem by using a cylinder of area A that only encloses a small portion of the sheet of charge (similar to every example online of this classic problem). Why does this small Area A represent the E field for the entire sheet? I'm also wondering why I cannot use a sphere and symmetry to solve this problem; Is the E field not constant?

## Homework Equations

electric flux = e * da;
flux = charge enclosed/epsilon nought
area of ends of a cylinder 2*pi*r^2

## The Attempt at a Solution

I have found the answer using a cylinder, but I don't see why this works, or why I can ignore part of the charge of the sheet; Or in other words, why charge enclosed in the above equation need not be equal to the charge of the sheet, but that of a tiny cross sectional area.

You can use just about any type of surface for this problem. A cylinder just works well. The crux of the problem is that the electric field is constant for the sheet, and to find it you examine just a small portion of area and a small portion of charge enclosed within that area, and figure out the field produced by that area. Then, because the field is constant, the value of the field you find there applies to the whole sheet.

The electric field reflects the symmetry of the charge distribution. An infinite sheet of charge is laterally homogeneous, the charge /unit area is the same everywhere: so must be the electric field. That means, the field can depend only on the distance from the sheet.
The charge distribution does not change if you rotate the sheet about an axis normal to the sheet: The electric field is also invariant for rotation about the same axis. It does not depend on direction. That means the electric field is normal to the sheet.

If there is nothing else but the sheet then the sheet is a mirror plane. The electric field at one side of the sheet is the mirror image of the electric field at the other side.

Knowing all these you can choose a Gaussian surface, which also has the symmetry of the situation: A right cylinder or right prism with opposite faces parallel with the sheet, the other face(s) normal to it. For such surface, the net flux is equal to the sum of fluxes across the parallel faces.

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## What is Gauss's Law applied to an infinite sheet of charge?

Gauss's Law is a fundamental principle in electromagnetism that describes the relationship between electric charges and electric fields. When applied to an infinite sheet of charge, Gauss's Law states that the electric flux through any closed surface surrounding the sheet is equal to the total charge enclosed by that surface divided by the permittivity of free space.

## How is Gauss's Law applied to an infinite sheet of charge used in physics?

Gauss's Law applied to an infinite sheet of charge is used in physics to calculate the electric field created by a charged sheet. This is useful in understanding the behavior of electric fields in various scenarios, such as in capacitors and parallel plate systems.

## What are the assumptions made when applying Gauss's Law to an infinite sheet of charge?

There are three main assumptions made when applying Gauss's Law to an infinite sheet of charge: 1) The sheet of charge is infinite in extent, meaning the electric field is constant at all points on the sheet. 2) The sheet of charge has a uniform charge density, meaning the charge is evenly distributed across the sheet. 3) The sheet of charge is flat, meaning there is no curvature or variation in the surface of the sheet.

## What is the formula for calculating the electric field using Gauss's Law applied to an infinite sheet of charge?

The formula for calculating the electric field using Gauss's Law applied to an infinite sheet of charge is E = σ/2ε0, where E is the electric field, σ is the charge density, and ε0 is the permittivity of free space.

## Can Gauss's Law applied to an infinite sheet of charge be used in real-world situations?

Yes, Gauss's Law applied to an infinite sheet of charge can be used in real-world situations. While an infinite sheet of charge does not exist in reality, it can be used as an approximation for systems with large, flat surfaces and uniform charge distributions. For example, it can be applied to parallel plate capacitors and the electric fields around large conducting plates.

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