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## Homework Statement

We have a coordinate system (x, y, z). Two conducting plates 1 and 2 are parallel and lie in the x-y plane. Plate 1 is at height (x, y, 0) and plate 2 is at height (x, y, 4a), where a is an arbitrary constant.

Between these two plates there are 2 charges +q and -q. Charge -q is at a point (0, 0, a), charge +q is at a point (0, 0, 3a).

What is the induced charge on plate 1 and 2?

## Homework Equations

[tex]Q=\int_{0}^{\infty }E\cdot \varepsilon _{0}2\pi rdr[/tex]

[tex]Q=\int_{0}^{\infty }E\cdot \varepsilon _{0}2\pi rdr[/tex]

3. The Attempt at a Solution [/B]

My first step was to discover how the charges were layed out. I think they are in a row of alternating signs at distance 2a one from another along the z axis.

After that my first method was to say that the induced charge is equal to sum of all image charges of the respective plate:

[tex] Q_{induced} =\sum_{n=0}^{\infty }(-1)^{n}Q=Q/2 [/tex]

My argument is the following:

[tex] Q_{induced} = +Q-Q+Q-Q+Q-\cdot \cdot \cdot [/tex](1)

[tex] Q_{induced} +Q=+Q+Q-Q+Q-Q+Q-\cdot \cdot \cdot [/tex] (2)

We have two possible outcomes, the sum is either 1 or 0, but if the last term is one,

I can argue that it is infinitely far away, thus the charge it induces is cca 0.

If this is so then we can add equations (1) and (2) which will yield the following result:

[tex]2Q_{induced}+Q=2Q[/tex]

[tex]Q_{induced}=Q/2[/tex]

This was the first method, second method was to consider a point at distance r from the

z axis at the height of 4a (This is where the plate is, I'm looking for a functional dependence of E field so that I can find charge per area).

After a bit of algebra I find this sum:

[tex]E=\sum_{n=0}^{\infty }\frac{(-1)^{n}2kQ(2n+1)a}{((2n+1)^{2}a^{2}+r^{2})^{3/2}}[/tex]

After I plug it in this equation:

[tex]Q=\int_{0}^{\infty }E\cdot \varepsilon _{0}2\pi rdr[/tex]

I find the following result:

[tex]Q_{induced}=Qa\sum_{n=0}^{\infty }(-1)^{n}(2n+1)\int_{0}^{\infty }\frac{rdr}{((2n+1)^{2}a^{2}+r^{2})^{3/2}}[/tex]

After integrating I find this result:

[tex]Q_{induced}=\sum_{n=0}^{\infty }(-1)^{n}Q[/tex]

This is very weird, the problem is that I'm new to physics and I don't have much mathematics under my belt to really confirm that this really works, but I know that induced charge on this plate must be between 0 and 1 and that electrostatics ( the name implies) is static i.e. nature doesn't do it two ways, so I came here for help; Thanks.

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