91Ó°ÊÓ

The radius if the cylinder is $R$.

The uniform surface charge in upper half of the cylinder is ${\mathrm{σ}}_0$.

The uniform surface charge in upper lower of the cylinder is $−{\mathrm{σ}}_0$.

The expression for the potential is given as follows.

$V(s,\mathrm{Ï•})=a_0+b_{0}lns+\underset{K=1}{\overset{\mathrm{Î\pm }}{∑}}[s_p^k(a_{k}cosk\mathrm{Ï•}+b_{k}sink\mathrm{Ï•})+s^{-k}(c_{k}cosk\mathrm{Ï•}+d_{k}sink\mathrm{Ï•})]$…… (1)

Here,$a_k$, $b_k$,$c_k$ and $d_k$are the constant.

From equation (1)

The potential inside the shell is given as follows

$V_{in}(s,\mathrm{Ï•})=\underset{K=1}{\overset{\mathrm{Î\pm }}{∑}}{s}^k(a_{k}cosk\mathrm{Ï•}+b_{k}sink\mathrm{Ï•})$ ……. (2)

The potential outside the shell is given as follows.

$V_{out}(s,\mathrm{Ï•})=\underset{K=1}{\overset{\mathrm{Î\pm }}{∑}}{s}^{-k}(c_{k}cosk\mathrm{Ï•}+d_{k}sink\mathrm{Ï•})$ …… (3)

At the boundary condition, potential is continuous at . $s=R$Hence, equate the potential inside and outside the cylinder.

$\underset{K=1}{\overset{\mathrm{Î\pm }}{∑}}{R}^k(a_{k}cosk\mathrm{Ï•}+b_{k}sink\mathrm{Ï•})=\underset{K=1}{\overset{\mathrm{Î\pm }}{∑}}{R}^{-k}(c_{k}cosk\mathrm{Ï•}+d_{k}sink\mathrm{Ï•})$

Compare the coefficient of$\mathrm{cosk}\mathrm{Ï•}$and$\sin k\mathrm{Ï•}$from the above equation.

$\begin{array}{c}{a}_k{R}^k=R^{−k}{c}_k\\ c_k=R^{2k}{a}_k\end{array}$

Similarly,

$\begin{array}{c}{b}_k{R}^k=R^{−k}{d}_k\\ d_k=R^{2k}{b}_k\end{array}$

Consider the equation which relates the normal derivative of the potential with the surface charge density.

${\frac{∂V}{∂s}|}_{R+}−{\frac{∂V}{∂s}|}_{R−}=\frac{−\mathrm{σ}}{{\mathrm{Î\mu }}_0}$ ……. (4)

Calculate the derivative of potential inside cylinder.

$\begin{array}{l}{\frac{∂V}{∂s}|}_{R+}=∑\frac{∂}{∂s}{s}^{-k}{(c_{k}cosk\mathrm{ϕ}+d_{k}sink\mathrm{ϕ})}_{s=R}\\ {\frac{∂V}{∂s}|}_{R+}=∑\left(\frac{−k}{s^{k+1}}\right){(c_{k}cosk\mathrm{ϕ}+d_{k}sink\mathrm{ϕ})}_{s=R}\\ {\frac{∂V}{∂s}|}_{R+}=∑\left(\frac{−k}{R^{k+1}}\right)(c_{k}cosk\mathrm{ϕ}+d_{k}sink\mathrm{ϕ})\end{array}$

Calculate the derivative of potential outside cylinder.

$\begin{array}{l}{\frac{∂V}{∂s}|}_{R−}=∑\frac{∂}{∂s}{s}^k{(a_{k}cosk\mathrm{ϕ}+b_{k}sink\mathrm{ϕ})}_{s=R}\\ {\frac{∂V}{∂s}|}_{R−}=∑\left(k{s}^{k−1}\right){(a_{k}cosk\mathrm{ϕ}+b_{k}sink\mathrm{ϕ})}_{s=R}\\ {\frac{∂V}{∂s}|}_{R−}=∑\left(k{R}^{k−1}\right)(a_{k}cosk\mathrm{ϕ}+b_{k}sink\mathrm{ϕ})\end{array}$

Recall the equation (4),

${\frac{∂V}{∂s}|}_{R+}−{\frac{∂V}{∂s}|}_{R−}=\frac{−\mathrm{σ}}{{\mathrm{Î\mu }}_0}$

Substitute $∑\left(\frac{−k}{R^{k+1}}\right)(c_{k}cosk\mathrm{ϕ}+d_{k}sink\mathrm{ϕ})$for${\frac{∂V}{∂s}|}_{R+}$and $∑\left(k{R}^{k−1}\right)(a_{k}cosk\mathrm{ϕ}+b_{k}sink\mathrm{ϕ})$for${\frac{∂V}{∂s}|}_{R−}$into above equation.

$∑\left(\frac{−k}{R^{k+1}}\right)(c_{k}cosk\mathrm{Ï•}+d_{k}sink\mathrm{Ï•})+∑\left(k{R}^{k−1}\right)(a_{k}cosk\mathrm{Ï•}+b_{k}sink\mathrm{Ï•})=\frac{−\mathrm{σ}}{{\mathrm{Î\mu }}_0}$

Substitute $R^{2k}{a}_k$for $c_k$and $R^{2k}{b}_k$for$d_k$into above equation.

$\begin{array}{l}∑\left(\frac{−k}{R^{k+1}}\right)(R^{2k}{a}_{k}cosk\mathrm{Ï•}+R^{2k}{b}_{k}sink\mathrm{Ï•})−∑\left(k{R}^{k−1}\right)(a_{k}cosk\mathrm{Ï•}+b_{k}sink\mathrm{Ï•})=\frac{−\mathrm{σ}}{{\mathrm{Î\mu }}_0}\\ −∑\left(k{R}^{k−1}\right)(a_{k}cosk\mathrm{Ï•}+b_{k}sink\mathrm{Ï•})−∑\left(k{R}^{k−1}\right)(a_{k}cosk\mathrm{Ï•}+b_{k}sink\mathrm{Ï•})=\frac{−\mathrm{σ}}{{\mathrm{Î\mu }}_0}\\ ∑2k{R}^{k−1}(a_{k}cosk\mathrm{Ï•}+b_{k}sink\mathrm{Ï•})=\frac{\mathrm{σ}}{{\mathrm{Î\mu }}_0}\end{array}$ …â¶Ä¦(5)

Noe defines the above equation for the intervals,

$∑2k{R}^{k−1}(a_{k}cosk\mathrm{Ï•}+b_{k}sink\mathrm{Ï•})=\{\begin{array}{l}\frac{{\mathrm{σ}}_0}{{\mathrm{Î\mu }}_0}\text{ â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰}(0<\mathrm{Ï•}<\mathrm{Ï€})\\ \frac{−{\mathrm{σ}}_0}{{\mathrm{Î\mu }}_0}\text{ â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰}(\mathrm{Ï€}<\mathrm{Ï•}<2\mathrm{Ï€})\end{array}$

The value of integral of above angles,

$\begin{array}{l}{∫}_0^{2\mathrm{Ï€}}\mathrm{sink}\mathrm{Ï•}\cos l\mathrm{Ï•}d\mathrm{Ï•}=0\\ {∫}_0^{2\mathrm{Ï€}}cosk\mathrm{Ï•}\cos l\mathrm{Ï•}d\mathrm{Ï•}=\{\begin{array}{l}0\text{ â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰}k≠l\\ \mathrm{Ï€}\text{ â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰}k=l\end{array}\text{ â¶Ä‰}\end{array}$

Multiply the equation (5) with $\mathrm{cosl}\mathrm{Ï•}$.

$\begin{array}{l}2k{R}^{k−1}[{∫}_0^{2\mathrm{Ï€}}{a}_k\cos k\mathrm{Ï•}\cos l\mathrm{Ï•}d\mathrm{Ï•}+{∫}_0^{2\mathrm{Ï€}}{b}_k\sin k\mathrm{Ï•}\mathrm{cosl}\mathrm{Ï•}d\mathrm{Ï•}]=\frac{{\mathrm{σ}}_0}{{\mathrm{Î\mu }}_0}{[{∫}_0^{2\mathrm{Ï€}}\cos l\mathrm{Ï•}d\mathrm{Ï•}−{∫}_0^{2\mathrm{Ï€}}\cos l\mathrm{Ï•}]}_{}\\ 2k{R}^{k−1}\mathrm{Ï€}{a}_k=\frac{{\mathrm{σ}}_0}{{\mathrm{Î\mu }}_0}{\left[\frac{\sin l\mathrm{Ï•}}{l}\right]}_0^{\mathrm{Ï€}}\\ 2k{R}^{k−1}\mathrm{Ï€}{a}_k=\frac{{\mathrm{σ}}_0}{{\mathrm{Î\mu }}_0}\left(0\right)\\ a_k=0\end{array}$

The value of$a_k$is becomes zero therefore multiply with $\sin l\mathrm{Ï•}$and integrate.

$\begin{array}{l}2k{R}^{k−1}[{∫}_0^{2\mathrm{Ï€}}{a}_k\cos k\mathrm{Ï•}\sin l\mathrm{Ï•}d\mathrm{Ï•}+{∫}_0^{2\mathrm{Ï€}}{b}_k\sin k\mathrm{Ï•}\mathrm{sinl}\mathrm{Ï•}d\mathrm{Ï•}]=\frac{{\mathrm{σ}}_0}{{\mathrm{Î\mu }}_0}[{∫}_0^{2\mathrm{Ï€}}\sin l\mathrm{Ï•}d\mathrm{Ï•}−{∫}_0^{2\mathrm{Ï€}}\cos l\mathrm{Ï•}]\\ 2k{R}^{k−1}\mathrm{Ï€}{b}_k=\frac{{\mathrm{σ}}_0}{{\mathrm{Î\mu }}_0}[{(−\frac{\cos l\mathrm{Ï•}}{l})}_0^{\mathrm{Ï€}}+{\left(\frac{\cos l\mathrm{Ï•}}{l}\right)}_0^{2\mathrm{Ï€}}]\\ b_k=\frac{{\mathrm{σ}}_0}{l{\mathrm{Î\mu }}_0}(2−2\cos l\mathrm{Ï€})\\ b_k=\frac{{\mathrm{σ}}_0}{2k\mathrm{Ï€}l{\mathrm{Î\mu }}_0{R}^{k−1}}(2−2\cos l\mathrm{Ï€})\end{array}$

The value of$\mathrm{Ï€}$ is obtained at the condition .$k=l$

$b_k=\text{ }\{\begin{array}{l}0\text{ â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â€‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰if }l\text{ i²õ e±¹±ð²Ô}\\ \frac{2{\mathrm{σ}}_0}{\mathrm{Ï€}{k}^2{R}^{k−1}{\mathrm{Î\mu }}_0}\text{ â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰if }l\text{ i²õ o»å»å}\end{array}$

Similarly for the outside of the cylinder.

$\begin{array}{l}2k{R}^{k−1}[{∫}_0^{2\mathrm{Ï€}}{c}_k\cos k\mathrm{Ï•}\sin l\mathrm{Ï•}d\mathrm{Ï•}+{∫}_0^{2\mathrm{Ï€}}{d}_k\sin k\mathrm{Ï•}\mathrm{sinl}\mathrm{Ï•}d\mathrm{Ï•}]=\frac{{\mathrm{σ}}_0}{{\mathrm{Î\mu }}_0}[{∫}_0^{2\mathrm{Ï€}}\sin l\mathrm{Ï•}d\mathrm{Ï•}−{∫}_0^{2\mathrm{Ï€}}\cos l\mathrm{Ï•}]\\ 2k{R}^{k−1}\mathrm{Ï€}{d}_k=\frac{{\mathrm{σ}}_0}{{\mathrm{Î\mu }}_0}[{(−\frac{\cos l\mathrm{Ï•}}{l})}_0^{\mathrm{Ï€}}+{\left(\frac{\cos l\mathrm{Ï•}}{l}\right)}_0^{2\mathrm{Ï€}}]\\ d_k=\frac{{\mathrm{σ}}_0}{l{\mathrm{Î\mu }}_0}(2−2\cos l\mathrm{Ï€})\\ d_k=\frac{{\mathrm{σ}}_0}{2k\mathrm{Ï€}l{\mathrm{Î\mu }}_0{R}^{k−1}}(2−2\cos l\mathrm{Ï€})\end{array}$

The value of$\mathrm{Ï€}$ is obtained at the condition $k=l$.

$d_k=\text{ }\{\begin{array}{l}0\text{ â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â€‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰if }l\text{ i²õ e±¹±ð²Ô}\\ \frac{2{\mathrm{σ}}_0}{\mathrm{Ï€}{k}^2{R}^{k−1}{\mathrm{Î\mu }}_0}\text{ â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰if }l\text{ i²õ o»å»å}\end{array}$

Substitute$\frac{2{\mathrm{σ}}_0}{\mathrm{Ï€}{k}^2{R}^{k−1}{\mathrm{Î\mu }}_0}$ for $b_k$and $0$for $a_k$into equation (2).

$\begin{array}{l}{V}_{in}(s,\mathrm{Ï•})=\underset{K=1}{\overset{\mathrm{Î\pm }}{∑}}{s}^k(\left(0\right)cosk\mathrm{Ï•}+\frac{2{\mathrm{σ}}_0}{\mathrm{Ï€}{k}^2{R}^{k−1}{\mathrm{Î\mu }}_0}sink\mathrm{Ï•})\\ V_{in}(s,\mathrm{Ï•})=\underset{K=1}{\overset{\mathrm{Î\pm }}{∑}}{s}^k\left(\frac{2{\mathrm{σ}}_0}{\mathrm{Ï€}{k}^2{R}^{k−1}{\mathrm{Î\mu }}_0}sink\mathrm{Ï•}\right)\\ V_{in}(s,\mathrm{Ï•})=\frac{2{\mathrm{σ}}_{0}R}{\mathrm{Ï€}{\mathrm{Î\mu }}_0}\underset{K=1,3,5}{∑}\frac{1}{k^2}\sin k\mathrm{Ï•}{\left(\frac{s}{R}\right)}^k\end{array}$

Substitute $\frac{2{\mathrm{σ}}_0}{\mathrm{Ï€}{k}^2{R}^{k−1}{\mathrm{Î\mu }}_0}$for $d_k$and$0$ for$c_k$ into equation (3).

$\begin{array}{l}{V}_{out}(s,\mathrm{Ï•})=\underset{K=1}{\overset{\mathrm{Î\pm }}{∑}}{s}^{-k}(\left(0\right)cosk\mathrm{Ï•}+\frac{2{\mathrm{σ}}_0}{\mathrm{Ï€}{k}^2{R}^{k−1}{\mathrm{Î\mu }}_0}sink\mathrm{Ï•})\\ V_{out}(s,\mathrm{Ï•})=\underset{K=1}{\overset{\mathrm{Î\pm }}{∑}}{s}^{-k}\left(\frac{2{\mathrm{σ}}_0}{\mathrm{Ï€}{k}^2{R}^{k−1}{\mathrm{Î\mu }}_0}sink\mathrm{Ï•}\right)\\ V_{out}(s,\mathrm{Ï•})=\frac{2{\mathrm{σ}}_{0}R}{\mathrm{Ï€}{\mathrm{Î\mu }}_0}\underset{K=1,3,5}{∑}\frac{1}{k^2}\sin k\mathrm{Ï•}{\left(\frac{R}{s}\right)}^k\end{array}$

Hence, the expression for the potential inside the cylinder is$\frac{2{\mathrm{σ}}_{0}R}{\mathrm{Ï€}{\mathrm{Î\mu }}_0}\underset{K=1,3,5}{∑}\frac{1}{k^2}\sin k\mathrm{Ï•}{\left(\frac{s}{R}\right)}^k$ and outside the cylinder is $\frac{2{\mathrm{σ}}_{0}R}{\mathrm{Ï€}{\mathrm{Î\mu }}_0}\underset{K=1,3,5}{∑}\frac{1}{k^2}\sin k\mathrm{Ï•}{\left(\frac{R}{s}\right)}^k$.