Electric Field and Charge Density

In summary: Therefore, the charge density is not constant and depends on the position x. In summary, we can use Gauss's Law to relate the electric field and charge density of an infinite slab with a non-constant charge density.
  • #1
jesuslovesu
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[SOLVED] Electric Field and Charge Density

Oops, nevermind I guess I just use div(E) = rho/e0

Homework Statement



A layer of charge fills the space between x = -a and x = a. The layer has a charge density [tex]\rho (x)[/tex]. The electric field intensity everywhere inside the charge distribution is given by [tex]E(x) = \hat{x} Kx^3[/tex] where K is a constant[/tex]

Homework Equations


The Attempt at a Solution



I asked my professor about this and he said the [tex]\rho(x)[/tex] should be a volume charge density. So basically it's an infinite slab (in the y and z dir) Having some difficulty in finding the charge density.

I am assuming the charge density is NOT constant everywhere, correct?
I recognize that this requires a Gauss's Law formulation. Similar to an infinite plane if I am not mistaken.
[tex]E(A) = Qin/e0[/tex]
[tex]E(2A) = \rho (x) * A * (2a)/e0[/tex]
[tex]E = \rho (x) * a/e0 [/tex]
Can I just plug in E and rearrange to get [tex]\rho (x)[/tex] ? Am I handling the ends correctly? I am basically following the same procedure for finding the E of an infinite plane except I am using 2a as the thickness.
 
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  • #2
SOLUTION:Yes, you can use Gauss's Law to solve this problem. First, we need to calculate the electric field at a given point x inside the charge distribution: E(x) = \hat{x} Kx^3Then, we can calculate the total charge inside the charge distribution by integrating the charge density over the region between x=-a and x=a: Q_{total} = \int_{-a}^{a} \rho(x) dxFinally, we can use Gauss's Law to relate the electric field and charge density: E(x) = \frac{Q_{total}}{A\epsilon_0}where A is the area of the charge distribution (in this case it is 2a). We can then rearrange this equation to solve for the charge density: \rho(x) = \frac{E(x)\epsilon_0}{A} = \frac{\hat{x}Kx^3\epsilon_0}{2a}
 
  • #3

Yes, you are correct in using Gauss's Law to find the charge density in this situation. The charge density in this case is not constant and varies with position, so you will need to integrate over the entire charge distribution to find the total charge.

Using Gauss's Law, we can set up a Gaussian surface with an area of 2a in the x direction. The electric field at this surface is given by E = \rho (x) * a/e0. Then, we can use the given electric field equation, E(x) = \hat{x} Kx^3, to solve for the charge density at this surface:

E = \rho (x) * a/e0
\hat{x} Kx^3 = \rho (x) * a/e0
\rho (x) = \hat{x} Kx^3 * e0/a

This gives us the charge density at the surface, but we need to find the total charge by integrating over the entire charge distribution. This can be done by setting up an integral from -a to a for the charge density, \rho (x), and then solving for the total charge, Q.

Q = \int_{-a}^{a} \rho (x) dx

I hope this helps! Let me know if you have any further questions.
 

Related to Electric Field and Charge Density

1. What is an electric field?

An electric field is a region in space where an electric charge experiences a force. It is represented by a vector quantity, with both magnitude and direction.

2. How is an electric field created?

An electric field is created by a charged object. Positive charges create an outward electric field, while negative charges create an inward electric field.

3. What is charge density?

Charge density is a measure of the amount of electric charge per unit volume or area. It is often represented by the symbol ρ (rho) and is typically measured in coulombs per cubic meter or coulombs per square meter.

4. How is charge density related to electric field?

The electric field is directly proportional to the charge density. This means that as the charge density increases, the electric field also increases.

5. How can electric field and charge density be calculated?

Electric field can be calculated using the formula E = kq/r^2, where k is the Coulomb's constant, q is the charge, and r is the distance from the charge. Charge density can be calculated by dividing the total charge by the volume or area in which it is contained.

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