Charge distribution in an electric field

[teme92]

Homework Statement

An infinitely long cylinder of radius R is placed above an infinite grounded plane. The centre of the cylinder is a distance (d) above the plane, and the cylinder has a surface charge density of σ.

(a) Initially ignoring the grounded plane, what is the electric field due to the charged cylinder, assuming a cylindrical coordinate system that is centred on the cylinder?

(b) Use your result to provide and equation for the electric field in the xyz coordinate system, where the grounded plane is at z=0, and the centre of the cylinder is along the y-direction at z=d?

(c) What is the Electric field due to the charged sphere and the grounded plane for z>0?

(d) What is the distribution of the charge that is generated on the infinite grounded plane?

(e) Calculate the charge per unit length that is generated on the grounded plane in the y-direction, and use this to show that charge is conserved.

Homework Equations

Gauss's Law: $\int E{\cdot}da = \frac{Q_{enc}}{\epsilon_0}$

The Attempt at a Solution

(a) I used Gauss' Law where $da = 2{\pi}RL$ and $Q_{enc} = \sigma$, simplifying down to get:

$E = \frac{{\sigma}R}{2\epsilon_0}$

(b) I simply said $R = \sqrt{x^2+d^2}$ and subbed into (a) and got:

$E = \frac{{\sigma}(\sqrt{x^2+d^2})}{2\epsilon_0}$

(c) I got the $R_+$ and $R_-$ and then their respective $E_+$ and $E_-$. Then added them together for total $E$. This is a long equation that I won't type out unless its relevant for later parts.

(d),(e) These parts are confusing me and I don't understand how to answer them. Any help in the right direction would be really appreciated. Thanks in advance.

 

[tommyxu3]

(a) I used Gauss' Law where $da = 2{\pi}RL$ and $Q_{enc} = \sigma$, simplifying down to get:

$E = \frac{{\sigma}R}{2\epsilon_0}$

There may be a trivial error here that $da$ must have to match a also little amount like $2\pi RdL.$

A hint based on my opinion: The method of image. Have you learned it?

 

[teme92]

It says for part (a) to ignore the grounded plane so would that not therefore mean $d$ is irrelevant?

 

[tommyxu3]

I'm sorry that I didn't explain what I typed clearly. I used $da$ and $dL$ to present little amount. These just can make the both sides of the equation equal.

 

[vela]

Homework Statement

An infinitely long cylinder of radius R is placed above an infinite grounded plane. The centre of the cylinder is a distance (d) above the plane, and the cylinder has a surface charge density of σ.

(a) Initially ignoring the grounded plane, what is the electric field due to the charged cylinder, assuming a cylindrical coordinate system that is centred on the cylinder?

(b) Use your result to provide and equation for the electric field in the xyz coordinate system, where the grounded plane is at z=0, and the centre of the cylinder is along the y-direction at z=d?

(c) What is the Electric field due to the charged sphere and the grounded plane for z>0?

(d) What is the distribution of the charge that is generated on the infinite grounded plane?

(e) Calculate the charge per unit length that is generated on the grounded plane in the y-direction, and use this to show that charge is conserved.

Homework Equations

Gauss's Law: $\int E{\cdot}da = \frac{Q_{enc}}{\epsilon_0}$

The Attempt at a Solution

(a) I used Gauss' Law where $da = 2{\pi}RL$ and $Q_{enc} = \sigma$, simplifying down to get:

$E = \frac{{\sigma}R}{2\epsilon_0}$

The equation $da = 2{\pi}RL$ isn't correct. The lefthand side is infinitesimal but the righthand side is finite. Writing $Q = \sigma$ doesn't make sense either. The units don't match.

Don't mix up the radius of the cylinder $R$ with the radial coordinate $r$. Also, to be complete, you should consider the regions inside the cylinder and outside the cylinder.

(b) I simply said $R = \sqrt{x^2+d^2}$ and subbed into (a) and got:

Why would the radius of the cylinder $R$ depend on $x$ and the distance $d$ from the plane?

$E = \frac{{\sigma}(\sqrt{x^2+d^2})}{2\epsilon_0}$

(c) I got the $R_+$ and $R_-$ and then their respective $E_+$ and $E_-$. Then added them together for total $E$. This is a long equation that I won't type out unless its relevant for later parts.

(d),(e) These parts are confusing me and I don't understand how to answer them. Any help in the right direction would be really appreciated. Thanks in advance.

 

[teme92]

Hey thanks for the replies guys, I got help of a class mate today so everything is good.