Simplify the expression -- Electric field in a capacitor

[skrat]

Homework Statement

Between two metal plates a metal strip is inserted as shown on the figure.

a) Calculate the electric potential anywhere inside the capacitor.
b) Simplifly $U(x,y)$ for $x>>a$.
c) Calculate the electric field in the middle of the capactior.

Homework Equations

The Attempt at a Solution

a) Ok, since we have no other charges in the capacitor, than $\nabla ^2 U=0$. In kartezian coordinates this means that $U(x,y)=\sum _m (A_me^{-k_mx}+B_me^{k_mx})(Csin(k_my)+Dcos(k_my))$.

Using boundary conditions
$U(0,y)=U_0$ and
$U(x,0)=0$ and
$U(x,a)=0$ we find out that

$U(x,y)=\sum _{n=1} ^{\infty}\frac{2U_0}{n\pi }(1-(-1)^n)e^{-\frac{n\pi}{a}x}\sin(\frac{n\pi }{a}y)$

b) The result is that out of the sum only $n=1$ survives, therefore $U(x,y)=\frac{4U_0}{\pi }e^{-\frac{\pi}{a}x}\sin(\frac{\pi }{a}y)$.

I am having some troubles understanding this. Could anybody please try to explain how do I get this result?

c)

$U(x,a/2)=\sum _{n=1} ^{\infty}\frac{2U_0}{n\pi }(1-(-1)^n)e^{-\frac{n\pi}{a}x}\sin(\frac{n\pi }{2})$

$\vec E=-\nabla U(x,a/2)=\sum _{n=1} ^{\infty}\frac{2U_0}{a}(1-(-1)^n)\sin(\frac{n\pi }{2})e^{-\frac{n\pi}{a}x}$

I think I should do something more with that last equation but I really don't know what or how.. :/

 

[SteamKing]

Homework Statement

Simplifly $U(x,y)=\sum _{n=1} ^{\infty}\frac{2U_0}{n\pi }(1-(-1)^n)e^{-\frac{n\pi}{a}x}\sin(\frac{n\pi }{a}y)$ for $x>>a$.

Homework Equations

The Attempt at a Solution

The result is that out of the sum only $n=1$ survives, therefore $U(x,y)=\frac{4U_0}{\pi }e^{-\frac{\pi}{a}x}\sin(\frac{\pi }{a}y)$.

I am having some troubles understanding this. Could anybody please try to explain how do I get this result?

ps: If you need the whole problem, just let me know, but it shouldn't play any role for this...

Consider the expression (1 - (-1)ⁿ). What happens when n is even? When n is odd?

 

[skrat]

Consider the expression (1 - (-1)ⁿ). What happens when n is even? When n is odd?

How is this related to $x>>a$?

ps: SteamKing, I edited the post and added some more questions.

 

[SteamKing]

How is this related to $x>>a$?

ps: SteamKing, I edited the post and added some more questions.

Well, if you had examined the expression, you would have seen that about half the terms drop out, due to the expression being 0 for n even.

 

[skrat]

Well, if you had examined the expression, you would have seen that about half the terms drop out, due to the expression being 0 for n even.

Yes, I agree. But this still doesn't explain why only $n=1$ is ok? What happens with $n=3$?
I apologize but I still can't see how is this related to $x>>a$.