Force acting on a dipole in non-uniform electric field.

[anirudhsharma1]

Homework Statement

Calculate the force acting on a dipole of dipole moment P due to a line charge of density λ
at a distance r from it??

Homework Equations

field due to a line charge= λ/2εr

The Attempt at a Solution

tried caculating force on each individual charge but i don't see how dipole moment should come in play here?

 

[TSny]

The Attempt at a Solution

tried caculating force on each individual charge but i don't see how dipole moment should come in play here?

This approach should lead to the answer. If you can show some of the details of your calculation, maybe we can see how the dipole moment will come in.

A better approach is to use energy concepts. Are you familiar with the expression for the potential energy of a dipole in an electric field? Do you know how to relate potential energy to force?

 

[Vibhor]

Hello TSny ,

What is the orientation of the dipole with respect to line charge in this problem ?

Is it perpendicular to the line charge such that the center of dipole is at a distance 'r' ? Am I interpreting it correctly ?

 

[Vibhor]

Or is it that the end closer to the line charge is at a distance 'r' ?

 

[TSny]

The OP was not clear on the orientation. So I guess that P is oriented parallel to the E field. I think r can be taken as the distance to the center of the dipole.

 

[Vibhor]

Sir,

If that is the case and if +q is closer to line charge ,then

Net force on dipole = $\frac{2kλ}{(r-a)}q - \frac{2kλ}{(r+a)}q$ = $-\frac{2kλ\vec{p}}{(r^2-a^2)}$

If I consider r>>a ,then net force = $-\frac{2kλ\vec{p}}{r^2}$

Have I done it correctly ?

 

[TSny]

Looks very good. I believe that's correct.

 

[Vibhor]

A better approach is to use energy concepts. Are you familiar with the expression for the potential energy of a dipole in an electric field? Do you know how to relate potential energy to force?

Here I have a doubt . First I will show the work .

Potential Energy of a dipole in Electric field is $U =-\vec{p} \cdot \vec{E}$ . Since $\vec{p}$ and $\vec{E}$ are oppositely aligned , U = pE .

$U =\frac{2kλ\vec{p}}{r}$

$F=-\frac{dU}{dr}$

$F=-\frac{2kλp}{r^2}$

Have I done it correctly ?

If you think I have done it correctly , my doubt is that even though electric field across the length of dipole is non uniform , still expression for potential energy of dipole remains $U =-\vec{p} \cdot \vec{E}$ .

But this expression for U was for uniform electric field . Can you explain it ?

Thanks .

 

[TSny]

I had in mind an "ideal dipole" where the length of the dipole is infinitesimally small. Then you can use $U = -\vec{p} \cdot \vec{E}$.

For a finite length you can still use potential energy. The potential energy of a point charge $q$ in the field of the line charge is $U = -2 k \lambda q \ln \frac{r}{r_0}$ where $r$ is the distance of $q$ from the line charge and $r_0$ is an arbitrarily chosen distance from the line charge for defining zero potiential.

So, the potential energy of the dipole (for the case where the +q is farther away) is $U = -2 k \lambda q \ln \frac{r+a}{r-a}$. Here, $r$ is the location of the center of the dipole.

You can then get the force from $F = -\frac {dU}{dr}$.

 

[Vibhor]

Thank you Sir .