Question

A point particle of charge Q is located at P along the axis of an electric dipole 1 at a distance r as shown in the figure. The point P is also on the equatorial plane of a second electric dipole 2 at a distance r. The dipoles are made of opposite charge q separated by a distance 2a. For the charge particle at P not to experience any net force, which of the following correctly describes the situation?

• A. The net force on $Q$ is zero if the two dipoles are arranged so that the field at $P$ due to dipole 1 (on its axis) is exactly cancelled by that due to dipole 2 (on its equatorial plane). In the dipole‐approximation this requires that $$\frac{2p}{r^3}=\frac{p}{r'^3}\quad\Longrightarrow\quad r'= \frac{r}{2^{1/3}},$$ with the dipole moments (or equivalently the placement of the positive and negative charges) chosen so that the two fields are oppositely directed.

Answer 1

Answer: (A)

The net force on $Q$ is zero if the two dipoles are arranged so that the field at $P$ due to dipole 1 (on its axis) is exactly cancelled by that due to dipole 2 (on its equatorial plane). In the dipole‐approximation this requires that

$$\frac{2p}{r^3}=\frac{p}{r'^3}\quad\Longrightarrow\quad r'= \frac{r}{2^{1/3}},$$

with the dipole moments (or equivalently the placement of the positive and negative charges) chosen so that the two fields are oppositely directed.

Answer 2

1. In the dipole approximation, the field on the axis is

   $E_{\rm axis}= \frac{2p}{4\pi\epsilon_0\,r^3}$

   and on the equatorial line it is

   $E_{\rm equ}= \frac{p}{4\pi\epsilon_0\,r^3}$.

2. For a charge $Q$ at $P$ to experience zero net force the two fields must be equal in magnitude and opposite in direction.

3. Since the axial field is twice the equatorial field at the same distance, one must change the distance so that

   $\frac{2p}{r^3}=\frac{p}{r'^3}$

   i.e.

   $r'=r/2^{1/3}$,

   or reverse the direction of one dipole.