Derive electric field of infinite plane from field of infinite line

[theneedtoknow]

Homework Statement

Approximate an infinite field of charge as a sheet of infinitely long charged wires each with charge dQ = λL . Use the formula for the electric field of a wire in the limit as L goes to infinity to derive the formula for an infinite sheet of charge dencity η. You need to express λ in terms of of the surface charge density η times an infinetismall distance.

Homework Equations

E of infinite line charge = λ/(4 π e0 r)

integral of x/(x^2+y^2) dy = arctan (y/x)

The Attempt at a Solution

Wow... I don't even know where to start
I'll assume the wires run parallel to the y-axis
then the area of each wire is L*dx (assumign dy is the thickness of each wire
η = Q / A Now the area of each wire is L*dx , and since the charge of each is dQ
η = dQ / L*dx = λL/L*dx = λ/dx

honestly I have no idea what else to do or if there was even a point to doing what i just did...help!

 

[Delphi51]

I think your formula relating the two charge densities is correct.
I would sketch some little circles side by side across the page to represent the cross sections of wires placed side by side. Pick a point P below that to be the point where you will calculate the E field due to the infinite number of wires. Draw a vertical line, length R from P to the nearest wire. Draw a line from P to any other wire and label that distance r, and call the distance along the wires from the center line x. Sketch the electric field due to the wire you chose and label it E = lambda/(4*pi*epsilon*r).

Now you must integrate over the line of wires (x) to get the sum of the electric fields due to all the wires. Note that by symmetry you only need concern yourself with the downward component of the electric field at P.

 

[theneedtoknow]

OK I think what you said helps a LOT! it has gotten me moving on this question
so λ = ηdx

the field due to the random wire chosen is ηdx / (4 π e0 r) in the radial direction, and r is root (Z^2 + x^2)
to get just the R component of it
its ηdx / (4 π e0 root (z^2 + x^2)) * Z / root (Z^2 + x^2) = η Z dx / (4 π e0 *(Z^2 + x^2))

so now if i take out η/4 π e0 i have left over (Z dx/ Z^2 + x^2) which is good cause that's the kind of integral which they give us the solution to as a hint
so am i now supposed to integrate this whole thing from 0 to x or what?

 

[Delphi51]

Great work! Note that you can take the z through the integral sign because it is a constant. Integrate from minus infinity to infinity - it is an infinite plane.

 

[theneedtoknow]

so i have η/4 π e0 * (arctan x/z | <from negative to positive inifnity which is η/4 π e0 * (Pi/2 + Pi/2) = η/4 π e0 * π = η/4e0 which is half of the actual field of an infintie plane which is η/2e0 so i must have missed a factor of 2 somewhere...what happened? Was i supposed to multiply the force in the Z direction by 2 since i was connecting my point P to 2 different wires (one directly above and one at an angle)?

 

[theneedtoknow]

or rather not really a wire directly above, but a wire in the other direction at the same angle as the first wire?

 

[Delphi51]

I also got η/4e0. I don't see the error.

a wire in the other direction at the same angle as the first wire

No, we took that into account by integrating from minus pi/2 to pi/2.

I'm taking a look at an old textbook (Halliday & Resnick) and it says the E field due to the line of charge is lambda over 2*pi*epsilon*r. Looks like the 4 we started with should have been a 2, and all our work is correct!

 

[theneedtoknow]

Hahaha oh man...my own notes on the infinite line field also say 2...I have nooo idea why i had typed 4 when i posted the question. Thank you so much for the help! I really appreciate it! :)

 

[Delphi51]

Most welcome - it was fun! You are so lucky to have these interesting questions to do.