Question

A dipole with an electric moment $\vec{p}$ , is located at a distance $\vec{r}$ from a long thread, charged uniformly with a linear density $λ$ . Find the force $\vec{F}$ acting on the dipole, if the vector $\vec{p}$ is oriented

(a) along the thread;
(b) along the radius vector $\vec{r}$
(c) at right angles to the thread and the radius vector $\vec{r}$ .

Accepted Answer

The direction of electric field due to the long wire is radially outward and perpendicular to the wire. Electric field due to a long uniformly charged thread or wire is given by,

$E_{r} = \frac{λ}{2 π ε_{0} r} \propto \frac{1}{r}, E_{θ} = 0$

and the force, $\vec{F} = p \frac{\partial \vec{E}}{\partial l} . . . (1)$

Here, $r$ is the perpendicular distance from the wire, for any point.

(a) $\vec{p}$ along the thread:

The electric field due to the thread, at a constant perpendicular distance from the thread, remains constant.

So, $\vec{E}$ does not change as the point of observation is moved along the thread.

From equation $(1)$ ,

$\vec{F} = 0$

(b) $\vec{p}$ along $\vec{r}$ :

$\vec{F} = F_{r} \vec{e_{r}} = \frac{λ p}{2 π ε_{0} r^{2}} \vec{e_{r}} = - \frac{λ \vec{p}}{2 π ε_{0} r^{2}}$ , (On using $\frac{\partial}{\partial r} \vec{e_{r}} = 0$ )

(c) $\vec{p}$ along $\vec{e_{θ}}$ :

$\vec{F} = p \frac{\partial}{r \partial θ} \frac{λ}{2 π ε_{0} r} e_{r}$

$= \frac{p λ}{2 π ε_{0} r^{2}} \frac{\partial \vec{e_{r}}}{\partial θ} = \frac{p λ}{2 π ε_{0} r^{2}} \vec{e_{θ}} = \frac{\vec{p} λ}{2 π ε_{0} r^{2}}$

I E Irodov Solutions for Chapter: ELECTRODYNAMICS, Exercise 1: CONSTANT ELECTRIC FIELD IN VACUUM

I E Irodov Physics Solutions for Exercise - I E Irodov Solutions for Chapter: ELECTRODYNAMICS, Exercise 1: CONSTANT ELECTRIC FIELD IN VACUUM

Questions from I E Irodov Solutions for Chapter: ELECTRODYNAMICS, Exercise 1: CONSTANT ELECTRIC FIELD IN VACUUM with Hints & Solutions

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