Potential Due to an Electric Dipole — Question 2

Question 2

A far-field angular expansion selects the component of the dipole moment

Charges $\pm q$ are at position vectors $\pm\mathbf d/2$. Let $\mathbf p=q\mathbf d$ and observe at $\mathbf r$ with $r\gg d$. Use infinity reference and the first-order expansion $(1+u)^{-1/2}\simeq1-u/2$.

1. Expand the two inverse distances and derive the dipole potential.

2. Find its values at polar angles 0, $\pi/2$ and $\pi$ relative to p, including signs.

Question 2 - Solution

A far-field angular expansion selects the component of the dipole moment

(a) Method and calculation

$$\frac1{|\mathbf r\mp\mathbf d/2|}\simeq\frac1r\left(1\pm\frac{\hat{\mathbf r}\cdot\mathbf d}{2r}\right).$$$$V\simeq\frac{k\mathbf p\cdot\hat{\mathbf r}}{r^2}=\frac{kp\cos\theta}{r^2}.$$

(b) Interpretation and further calculation

$$V(0)=kp/r^2,\quad V(\pi/2)=0,\quad V(\pi)=-kp/r^2.$$$$V(r,\pi-\theta)=-V(r,\theta).$$

Physical check

The monopole terms cancel because the total charge is zero; the first surviving term is the dipole projection.