The Electric Field Due to an Electric Dipole — Question 5

Question 5

Angular dipole components determine a field direction distinct from the observation ray

A point-dipole approximation has moment $p\hat{\mathbf z}$. At polar angle $\theta$, use $E_r=2A\cos\theta$, $E_\theta=A\sin\theta$, with $A=kp/r^3>0$ and $\hat{\boldsymbol\theta}$ pointing toward increasing polar angle.

1. At $\theta=60^\circ$, find $E_r,E_\theta$, the magnitude and the angle of E from the outward radial direction.

2. Find the z component using $\hat{\mathbf z}=\cos\theta\hat{\mathbf r}-\sin\theta\hat{\boldsymbol\theta}$.

Question 5 - Solution

Angular dipole components determine a field direction distinct from the observation ray

(a) Method and calculation

$$E_r=A,\quad E_\theta=\tfrac{\sqrt3}2A,\quad |\mathbf E|=\tfrac{\sqrt7}2A.$$$$\alpha=\tan^{-1}(E_\theta/E_r)=\tan^{-1}(\sqrt3/2)\approx40.8934^\circ.$$

(b) Interpretation and further calculation

$$E_z=E_r\cos\theta-E_\theta\sin\theta=A(3\cos^2\theta-1).$$$$\theta=60^\circ:\quad E_z=-A/4.$$A positive radial component does not imply a positive component along the dipole axis.

Physical check

The radial basis rotates with observation direction; it is not the fixed z basis.