91Ó°ÊÓ

The dipole moment is the product of charge and distance of separation charge from the origin.

Write the expression for dipole moment.

$\mathit{p}\mathbf{=}\underset{\mathbf{i}\mathbf{=}\mathbf{1}}{\overset{\mathbf{n}}{\mathbf{∑}}}{\mathit{q}}_{\mathbf{i}}{\mathit{r}}_{\mathbf{i}}$ ……. (1)

Here, is the charge of $i^{th}$particle,$r_i$ is the distance of separation charge form the origin.

The position of particle is shown in figure.

It consist of charges as q ,3q-2q,-2q. The separation between the charges denoted by a.

Write the expression for the total dipole moment due to four charges.

$\begin{array}{l}P=\left(3qa\hat{z}-qa\hat{z}\right)+\left(-2qa\hat{y}+-2qa\left(-\hat{y}\right)\right)\\ =2qa\hat{z}+\left(2qa\hat{y}\right)+2qa\hat{y}.........\left(2\right)\\ =2qa\hat{z}\end{array}$

Write the expression for the position of the point in spherical coordinates.

$$\begin{gathered} x=r\sin \mathrm{θ}\cos \mathrm{θ} \\ q=r\sin \mathrm{θ}\sin \mathrm{θ} \\ z=r\cos \mathrm{θ} \end{gathered}$$

The scalar product of dipole P and position vector $\hat{r}$is,

$P.\hat{r}$ ......(3)

Substitute $2qa\hat{z}$ for P in equation (3).

$$\begin{gathered} P.\hat{r}=2qa\hat{z}.\hat{r} \\ =2qa\left(\hat{z}.\hat{r}\right) \\ =2qa\left(\hat{r}\cos \mathrm{θ}.\hat{r}\right) \end{gathered}$$

Substitute $\hat{r}.\hat{r}=1$

Thus,

$P.\hat{r}=2qa\cos \mathrm{θ}.......\left(4\right)$

Write the equation for potential at distance rdue to point charge q.

$V=\frac{1}{4\mathrm{Ï€}{\mathrm{Î\mu }}_0}\frac{P.\hat{r}}{r^2}......\left(5\right)$

Substitute 2qacos$\mathrm{θ}$ for P .$\hat{r}$, in equation (5)

$V=\frac{1}{4\mathrm{Ï€}{\mathrm{Î\mu }}_0}\frac{2qa\cos \mathrm{θ}}{r^2}$

Thus, the potential at the origin for the point charge is $V=\frac{1}{4\mathrm{Ï€}\mathrm{Î\mu }}\frac{2qa\cos \mathrm{θ}}{r^2}$.