Potential Due to an Electric Dipole — Question 1

Question 1

The exact axial dipole voltage gives a controlled far-field error

A dipole consists of +q at z=d/2 and -q at z=-d/2, with q,d>0 and moment p=qd. Observe on the positive axis at z>d/2. Use infinity reference.

1. Find the exact voltage and its leading far-field approximation.

2. Find the relative error of the approximation using the exact voltage as denominator, and the condition for error below one percent.

Question 1 - Solution

The exact axial dipole voltage gives a controlled far-field error

(a) Method and calculation

$$V=kq\left(\frac1{z-d/2}-\frac1{z+d/2}\right)=\frac{kp}{z^2-d^2/4}.$$$$V_{\mathrm{far}}=kp/z^2\quad(z\gg d).$$

(b) Interpretation and further calculation

$$\frac{V_{\mathrm{far}}-V}{V}=-\frac{d^2}{4z^2}.$$$$\frac{d^2}{4z^2}<.01\Rightarrow z>5d.$$

Physical check

The dipole approximation underestimates the positive axial voltage because the exact denominator is smaller.