91Ó°ÊÓ

Write the expression for the potential $\mathit{V}\left(x,y\right)$in the infinite slot.

$\mathit{V}\left(x,y\right)\mathbf{=}\frac{\mathbf{4}{\mathbf{V}}_{\mathbf{0}}}{\mathbf{Ï€}}\underset{\mathbf{n}\mathbf{=}\mathbf{1}\mathbf{,}\mathbf{3}\mathbf{,}\mathbf{5}\mathbf{.}\mathbf{.}\mathbf{.}}{\mathbf{∑}}\frac{\mathbf{1}}{\mathbf{n}}{\mathit{e}}^{\frac{\mathbf{-}\mathbf{n}\mathbf{Ï€³\ae }}{\mathbf{a}}}\mathit{s}\mathit{i}\mathit{n}\left(\frac{n\mathrm{Ï€}y}{a}\right)$ …… (1)

Here, $V_0$is the constant potential along the conductor, $x$is the x-coordinate, $y$is the y-coordinate, and $n$is the positive integer.

Derive the charge density in terms of electric potential.

$$\begin{gathered} \mathrm{σ}=-e_0\frac{∂V}{∂n} \\ \mathrm{σ}\left(y\right)=-e_0{\left[\frac{∂V}{∂x}\right]}_{x=0} \end{gathered}$$ …… (2)

Substitute $\frac{4V_0}{\mathrm{Ï€}}\underset{n=1,3,5...}{∑}\frac{1}{n}{e}^{\frac{-n\mathrm{Ï€³\ae }}{a}}\sin \frac{n\mathrm{Ï€²â}}{a}$for $V\left(x,y\right)$in equation (2).

$$\begin{gathered} {\overline{)\mathrm{σ}\left(y\right)={\mathrm{Î\mu }}_0\frac{∂}{∂x}\left\{\frac{4V_0}{\mathrm{Ï€}}\underset{}{∑\frac{1}{n}{e}^{\frac{n\mathrm{Ï€³\ae }}{a}}}\sin \left(\frac{n\mathrm{Ï€²â}}{a}\right)\right\}}}_{x=0} \\ {\overline{)={\mathrm{Î\mu }}_0\frac{4V_0}{\mathrm{Ï€}}\frac{∂}{∂x}\left\{\underset{}{∑\frac{1}{n}{e}^{\frac{n\mathrm{Ï€³\ae }}{a}}}\sin \left(\frac{n\mathrm{Ï€²â}}{a}\right)\right\}}}_{x=0} \\ {\overline{)={\mathrm{Î\mu }}_0\frac{4V_0}{\mathrm{Ï€}}∑\frac{1}{n}\left\{\left(-\frac{nx}{a}\right)\underset{}{e^{\frac{n\mathrm{Ï€³\ae }}{a}}}\sin \left(\frac{n\mathrm{Ï€²â}}{a}\right)\right\}}}_{x=0} \\ {\overline{)={\mathrm{Î\mu }}_0\frac{4V_0}{\mathrm{Ï€}}\frac{1}{n}\left(\frac{nx}{a}\right)\underset{}{∑\frac{1}{n}{e}^{\frac{n\mathrm{Ï€³\ae }}{a}}}\sin \left(\frac{n\mathrm{Ï€²â}}{a}\right)}}_{x=0} \end{gathered}$$

$\mathrm{σ}\left(y\right)=\frac{4{\mathrm{Î\mu }}_0{V}_0}{a}\underset{n=1,3,5...}{∑}\sin \left(\frac{n\mathrm{Ï€²â}}{a}\right)$

Hence, the equation for the charge density on the strip at $x=0$is $\mathrm{σ}\left(y\right)=\frac{4{\mathrm{Î\mu }}_0{V}_0}{a}\underset{n=1,3,5...}{∑}\sin \left(\frac{n\mathrm{Ï€²â}}{a}\right)$.