Spherical Configuration of Charges

You are given three charges two of them equal with $+q$ situated on the sphere of radius $a$ at $(theta,phi) =(pi/4, 0)$ and $(pi/4, pi)$ and the third equal with $-q$ situated at $(pi,0)$. Please find the potential and electric field far from the origin.

Sphere Charges Configuration

Figure is above. if $a$ is the radius of the sphere and $r>>a$ is the distance to the point P where the potential is measured then

-The monopole term in the potential is


-The dipole term in the potential is

$V_{dip}(overrightarrow{r})=frac{1}{4piepsilon_0}frac{overrightarrow{p}hat{r}}{r^2}$ where $overrightarrow{p}=sum_i (q_i*overrightarrow{r_i})$

Since $overrightarrow{r_1}=-ahat k$, $overrightarrow{r_2}=acos(45)hat i+asin(45)hat k$ and $overrightarrow{r_3}=-acos(45)hat i+asin(45)hat k$ one has

$overrightarrow{p}=-q(a*hat k)+2q(asin(45)hat k)=qa(sqrt{2}+1)hat k$

so that

$V_{dip}(overrightarrow{r})=frac{qa(sqrt{2}+1)}{4piepsilon_0}*frac{hat rhat k}{r^2}=frac{qa(sqrt{2}+1}{4piepsilon_0 r^2}cos (theta)$

$V(r,theta,phi)=frac{q}{4piepsilon_0 r}left (1+frac{a(sqrt{2}+1)}{r}cos(theta)right )$

The electric field is

$overrightarrow{E}(r,theta,phi)=-nabla V=-(frac{dV}{dr}hat r-(frac{1}{r})(frac{dV}{dtheta})hat theta=frac{kq}{r^2}hat r-frac{2kqa(sqrt{2}+1)}{r^4}sin (theta)hat theta$