Answer to Question #231927 in Physics for Mosses07

Question

A dipole is centered at the origin of a coordinate system, and a small charged sphere is some distance away along the perpendicular bisector of the dipole. The particle carries a uniformly distributed charge of -3.0 nC, and experiences a 200 nN electric force in the positive y direction.

(a) If the dipole charge is 10 nC and the dipole separation is 20 mm, how far away from the dipole is the sphere?

(b) How is the dipole moment oriented?

Answer 1

Find the distance from the dipole ends to the sphere:

$$D^2=a^2+x^2$$

The force between the green and blue charge and the sphere:

$$F_1=k\frac{qQ}{D^2}=k\frac{qQ}{a^2+x^2}.\\\space\\ F_2=k\frac{qQ}{D^2}=k\frac{qQ}{a^2+x^2}.$$

The sine of the angle between the x-axis and the forces is

$$\sin\theta=\frac{a}{D}=\frac{a}{\sqrt{x^2+a^2}}.$$

The resultant of 200 nN is

$$R=F_1\sin\theta+F_2\sin\theta=\\\space\\ =\frac{2kqQ}{a^2+x^2}·\frac{a}{\sqrt{x^2+a^2}}=\frac{2kaqQ}{(a^2+x^2)^{\frac32}}.$$

Solve this equation using any method you prefer (I used Wolframalpha.com), the distance is

$$a=0.3\text{ m}.$$

As we see, the moment is oriented clockwise.