The Electric Field Due to an Electric Dipole — Question 6

Question 6

A zero axial component occurs on a cone where the field remains nonzero

For a point dipole $p\hat{\mathbf z}$, use $E_z=A(3\cos^2\theta-1)$ and $|\mathbf E|=A\sqrt{1+3\cos^2\theta}$, with $A=kp/r^3>0$.

1. Find both polar angles where the field has zero z component.

2. Find the magnitude there and decide whether either cone is a surface of field nulls.

Question 6 - Solution

A zero axial component occurs on a cone where the field remains nonzero

(a) Method and calculation

$$E_z=0\Rightarrow\cos^2\theta=1/3.$$$$\theta_1=\cos^{-1}(1/\sqrt3)\approx54.7356^\circ,\quad\theta_2=180^\circ-\theta_1\approx125.2644^\circ.$$

(b) Interpretation and further calculation

$$|\mathbf E|=A\sqrt{1+3(1/3)}=\sqrt2A>0.$$$$|\mathbf E|\ge A\quad\text{for every angle at finite }r.$$Only one component vanishes; a nonzero ideal dipole has no finite off-source field null.

Physical check

The two dashed cones are shown in cross-section; a component condition does not determine a zero vector.