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

1. May 27, 2013

### anirudhsharma1

1. The problem statement, all variables and given/known data
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??

2. Relevant equations
field due to a line charge= λ/2εr

3. The attempt at a solution
tried caculating force on each individual charge but i dont see how dipole moment should come in play here?

2. May 27, 2013

### TSny

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?

3. Jun 2, 2015

### 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 ?

Last edited: Jun 2, 2015
4. Jun 2, 2015

### Vibhor

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

Last edited: Jun 2, 2015
5. Jun 2, 2015

### 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.

6. Jun 2, 2015

### 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 ?

7. Jun 2, 2015

### TSny

Looks very good. I believe that's correct.

8. Jun 2, 2015

### Vibhor

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 .

Last edited: Jun 2, 2015
9. Jun 2, 2015

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

10. Jun 2, 2015

### Vibhor

Thank you Sir .

Last edited: Jun 3, 2015