91Ó°ÊÓ

The location of charge distribution and sphere is shown in below figure.

Here, P is the point at which electric field to be determined and z,r are the distances.

Write the expression for the dipole.

$\mathit{p}\mathbf{=}\mathit{r}\mathbf{\text{'}}\mathbf{∫}\left(a"\right)\mathit{Î\P }\mathbf{\text{'}}$ …… (1)

Here, r' is the distance, $\mathrm{ÒÏ}\left(r\text{'}\right)$is the charge density.

Diploes always point the positive charge and thus substitute z for r', ${\mathrm{ÒÏ}}_0$for $\mathrm{ÒÏ}\left(r\text{'}\right)$and $r^2\sin \mathrm{θ}drd\mathrm{θ}d\mathrm{Ï•}$for $d\mathrm{Î\P }$in equation (1),

$p=∫z{\mathrm{ÒÏ}}_0{r}^2\sin \mathrm{θ}drd\mathrm{θ}d\mathrm{Ï•}$ …… (2)

Determine value of z in terms of r.

$$\begin{gathered} \cos \mathrm{θ}=\frac{z}{2r} \\ z=2r\cos \mathrm{θ} \end{gathered}$$

Now, substitute $2r\cos \mathrm{θ}$for z.

To solve the integration, take the limits $\mathrm{θ}$from 0 to $\frac{\mathrm{π}}{2}$, r from 0 to R and from 0 to $2\mathrm{π}$.

$$\begin{gathered} p=2{\mathrm{ÒÏ}}_0\underset{0}{\overset{R}{∫}}{r}^{3}dr\underset{0}{\overset{\frac{\mathrm{Ï€}}{2}}{∫}}\cos \mathrm{θ}\sin \mathrm{θ}d\mathrm{θ}\underset{0}{\overset{2\mathrm{Ï€}}{∫}}d\mathrm{Ï•} \\ =2{\mathrm{ÒÏ}}_0{\left[\frac{r^4}{4}\right]}_0^R{\left[\frac{-\cos 2\mathrm{θ}}{4}\right]}_0^{\frac{\mathrm{Ï€}}{2}}{\left[\mathrm{Ï•}\right]}_0^{2\mathrm{Ï€}} \\ =-\frac{R^4}{4}\left(\frac{-1-1}{4}\right)\left(2\mathrm{Ï€}\right) \\ =\frac{{\mathrm{ÒÏ}}_0{R}^4\mathrm{Ï€}}{2} \end{gathered}$$

Now, write the formula for electric field in terms of dipole.

$E_{dipole}\left(r,\mathrm{θ}\right)=\frac{p}{4\mathrm{Ï€}{\mathrm{Î\mu }}_0{r}^3}\left(2\cos \widehat{\mathrm{θ}}r+\sin \widehat{\mathrm{θ}}\mathrm{θ}\right)$

Substitute $\frac{{\mathrm{ÒÏ}}_0{R}^4\mathrm{Ï€}}{2}$for $\mathrm{ÒÏ}$in above equation.

$$\begin{gathered} E_{dipole}\left(r,\mathrm{θ}\right)=\frac{\left({\displaystyle \frac{{\mathrm{ÒÏ}}_0{R}^4\mathrm{Ï€}}{2}}\right)}{4\mathrm{Ï€}{\mathrm{Î\mu }}_0{r}^3}\left(2\cos \widehat{\mathrm{θ}}+\sin \widehat{\mathrm{θ}}\mathrm{θ}\right) \\ =\frac{{\mathrm{ÒÏ}}_0{R}^4}{8\mathrm{Ï€}{\mathrm{Î\mu }}_0{r}^3}\left(2\cos \widehat{\mathrm{θ}}r+\sin \widehat{\mathrm{θ}}\mathrm{θ}\right) \end{gathered}$$

Hence, the magnitude of electric field is $\frac{{\mathrm{ÒÏ}}_0{R}^4}{8{\mathrm{Î\mu }}_0{r}^3}\left(2\cos \widehat{\mathrm{θ}}r+\sin \widehat{\mathrm{θ}}\mathrm{θ}\right)$.