91Ó°ÊÓ

A point charge Qis fixed on a table.

With Q at the center, a dipole $\stackrel{→}{p}$moves on a frictionless circular track of radius R,

and constrained always to point tangent to the circle.

The force on a dipole having moment $\stackrel{→}{p}$in the presence of an electric field $\stackrel{→}{E}$is

$\stackrel{\mathbf{→}}{\mathbf{F}}\mathbf{=}\left(\stackrel{→}{p}.\stackrel{→}{∇}\right)\stackrel{\mathbf{→}}{\mathbf{E}}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{\left(}\mathbf{1}\mathbf{\right)}$

Consider cylindrical coordinates.

The expression for the electric field from a point charge Q is

$\stackrel{→}{E}=\frac{Q}{4\mathrm{Ï€}{\mathrm{Î\mu }}_0{s}^2}\hat{s}$

Here, ${\mathrm{Î\mu }}_0$is the permittivity of free space.

Using equation (1), Force on the dipole $\stackrel{→}{p}$ moving on a circular track with Q at its center is

$$\begin{gathered} \stackrel{→}{F}=\frac{p}{s}\frac{∂}{∂\mathrm{θ}}\frac{Q}{4\mathrm{Ï€}{\mathrm{Î\mu }}_0{s}^2}\hat{s} \\ =\frac{p}{s}\frac{Q}{4\mathrm{Ï€}{\mathrm{Î\mu }}_0{s}^2}\frac{∂\hat{s}}{∂\mathrm{θ}} \\ =\frac{p}{s}\frac{Q}{4\mathrm{Ï€}{\mathrm{Î\mu }}_0{s}^2}\widehat{\mathrm{θ}} \\ =\frac{Q}{4\mathrm{Ï€}{\mathrm{Î\mu }}_0{s}^3}\stackrel{→}{p} \end{gathered}$$

The direction of the force is different for the positive and negative ends of the dipole. The net force acts tangential.