91Ó°ÊÓ

Write the expression for charge for the small line segment.

$\mathrm{ÒÏ}d\mathrm{Ï„}$or $\mathrm{λ}dz$ …… (1)

Here,P is the volume charge density and is the linear charge density.

Here, the charge Q is uniformly distributed along the z-axis, from$z=-atoz=+a$

$\mathrm{λ}=\frac{Q}{2a}$ …… (2)

Multiply with on both sides of the above equation.

$\mathrm{λ}dz=\frac{Q}{2a}dz$ ……. (3)

$V\left(r\right)=\frac{1}{4\mathrm{Ï€}{\mathrm{Î\mu }}_0}\underset{n=0}{\overset{∞}{∑}}\frac{1}{r^{n+1}}\underset{-a}{\overset{+a}{∫}}{z}^n{P}_n\left(\cos \mathrm{θ}\right)\frac{Q}{2a}dz$...... (4)

Now, take the following equation from equation (4)

$$\begin{gathered} \underset{-a}{\overset{+a}{∫}}{z^{n}dz=\left[\frac{z^{n+1}}{n+1}\right]}_{-a}^{+a} \\ =\frac{2a^{n+1}}{n+1} \end{gathered}$$ ……. (5)

If the n is even, then equation (5) can be as follows,

$$\begin{gathered} \underset{-a}{\overset{+a}{∫}}{z}^{n}dz=\frac{2a^{n+1}}{n+1} \\ =0 \end{gathered}$$

Substitute the equation (4) in (5)

$\begin{array}{rcl}V& =& \frac{1}{4\mathrm{Ï€}{\mathrm{Î\mu }}_0}\underset{0,2,4}{\overset{∞}{∑}}\frac{1}{r^{n+1}}\left(\frac{Q}{2a}\frac{2a^{n+1}}{n+1}\right)P_n\cos \mathrm{θ}\\ V\left(r\right)& =& \frac{1}{4\mathrm{Ï€}{\mathrm{Î\mu }}_0}\frac{Q}{r}\underset{0,2,4}{\overset{∞}{∑}}\left(\frac{1}{n+1}{\left(\frac{a}{r}\right)}^n{P}_n\cos \mathrm{θ}\right)\\ V\left(r,\mathrm{θ}\right)& =& \frac{1}{4\mathrm{Ï€}{\mathrm{Î\mu }}_0}\frac{Q}{r}\left[1+{\left(\frac{a}{r}\right)}^0{P}_0\cos \mathrm{θ}+\frac{1}{3}{\left(\frac{a}{r}\right)}^2{P}_2\cos \mathrm{θ}+\frac{1}{5}{\left(\frac{a}{r}\right)}^4{P}_4\cos \mathrm{θ}+....\right]\\ & =& \frac{Q}{4\mathrm{Ï€}{\mathrm{Î\mu }}_{0}r}\left[1+\frac{1}{3}{\left(\frac{a}{r}\right)}^2{P}_2\cos \mathrm{θ}+\frac{1}{5}{\left(\frac{a}{r}\right)}^4{P}_4\cos \mathrm{θ}+....\right]\\ & & \end{array}$

Therefore, the electric potential at a point is the proved. That is,

$\frac{Q}{4\mathrm{Ï€}{\mathrm{Î\mu }}_{0}r}\left[1+\frac{1}{3}{\left(\frac{a}{r}\right)}^2{P}_2\cos \mathrm{θ}+\frac{1}{5}{\left(\frac{a}{r}\right)}^4{P}_4\cos \mathrm{θ}+....\right]$