Two infinite parallel grounded conducting planes are held a distance a apart. A point charge q is placed in the region between them, a distance x from one plate. Find the force on q.^20 Check that your answer is correct for the special cases a → ∞ and x = a/2.

Question

Two infinite parallel grounded conducting planes are held a distance a apart. A point charge q is placed in the region between them, a distance x from one plate. Find the force on [latex]q .^{20}[/latex] Check that your answer is correct for the special cases a → ∞ and x = a/2.

Accepted Answer

The image configuration is shown in the figure; the positive image charge forces cancel in pairs. The net force of the negative image charges is:

[latex]F=\frac{1}{4 \pi \epsilon_{0}} q^{2}\left\{\frac{1}{[2(a-x)]^{2}}+\frac{1}{[2 a+2(a-x)]^{2}}+\frac{1}{[4 a+2(a-x)]^{2}}+\ldots ight.[/latex]

[latex]\left.-\frac{1}{(2 x)^{2}}-\frac{1}{(2 a+2 x)^{2}}-\frac{1}{(4 a+2 x)^{2}}-\ldots ight\}[/latex]

[latex]=\frac{1}{4 \pi \epsilon_{0}} \frac{q^{2}}{4}\left\{\left[\frac{1}{(a-x)^{2}}+\frac{1}{(2 a-x)^{2}}+\frac{1}{(3 a-x)^{2}}+\ldots ight]-\left[\frac{1}{x^{2}}+\frac{1}{(a+x)^{2}}+\frac{1}{(2 a+x)^{2}}+\ldots ight] ight\}[/latex].

When [latex]a ightarrow \infty ext { (i.e. } a \gg x) ext { only the } \frac{1}{x^{2}} ext { term survives: } F=-\frac{1}{4 \pi \epsilon_{0}} \frac{q^{2}}{(2 x)^{2}}[/latex] (same as for only one plane— Eq. 3.12). When x = a/2,

[latex]F =-\frac{1}{4 \pi \epsilon_{0}} \frac{q^{2}}{(2 d)^{2}} \hat{ z }[/latex] (3.12)

[latex]F=\frac{1}{4 \pi \epsilon_{0}} \frac{q^{2}}{4}\left\{\left[\frac{1}{(a / 2)^{2}}+\frac{1}{(3 a / 2)^{2}}+\frac{1}{(5 a / 2)^{2}}+\ldots ight]-\left[\frac{1}{(a / 2)^{2}}+\frac{1}{(3 a / 2)^{2}}+\frac{1}{(5 a / 2)^{2}}+\ldots ight] ight\}=0[/latex].

Question 3.39: Two infinite parallel grounded conducting planes are held a ...

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