Question

Two dipoles have dipole moment p1 and p2. Given p1 parallel to p2. How do you find the force between the dipoles if x is the distance between them?

Answer 1

If they are equal in size and direction then they would apply a force to each other in outward. If they are parallel but in opposite direction, this time, they would apply a force to each other inward. You should consider this problem as a square with 4 charges in tips.

Answer 2

Thank you :)

Answer 3

Two methods :

You can realise the dipoles as 4 charges and work out the force on either of them, then assume distance between charges is much less than distance between dipoles.

You can use expression of electric field of dipole no1 in its equatorial plane, then apply it to dipole no2 as \(\vec F=\vec \nabla(\vec P_2.\vec E_1)\). Note that when deriving, you will use distance between dipoles as variable.

Answer 4

Do you have the value of the answer?

Answer 5

Yep

\[\large \dfrac{1}{4 \pi \in_0} \dfrac{3p_1p_2}{r^4}\]

Answer 6

Correct!
Second method is faster, but you have to know the expression of net force acting on a dipole.

Answer 7

You mean \(\large E=\dfrac{1}{4 \pi \in_0}\dfrac{p}{r^3}\) ?

Answer 8

No, I mean:
\(\vec F=-\vec \nabla(\text{Potential Energy of dipole in external field})=-\vec \nabla(-\vec P_2.\vec E_1)=+\vec \nabla(\vec P_2.\vec E_1)\)

Answer 9

What does the inverted triangle represent? Derivative?

Answer 10

gradient operator. Here it represents \(\partial/\partial r\) because of symmetries.

Answer 11

Will be back in an hour's time.

Answer 12

ok

Answer 13

So \[\large E1=\dfrac{1}{4 \pi \in_0}\dfrac{p}{r^3}\] isnt it?

Answer 14

COrrect.
Actually, if \(\vec P_1=p_1\hat z\), then \(\vec E1=-\dfrac{1}{4 \pi \in_0}\dfrac{p_1}{r^3}\hat z\)

Suppose \(\vec P_2=p_2\hat z\)

Then: PEnergy = \(-\vec P_2.\vec E_1=+\dfrac{1}{4 \pi \in_0}\dfrac{p_1.p_2}{r^3}\)

Then net force = \(-\vec{grad}\;(\text{PEnergy})=-\dfrac{p_1.p_2}{4 \pi \in_0}\dfrac{\partial}{\partial r}(\dfrac{1}{r^3})=-\dfrac{p_1.p_2}{4 \pi \in_0}\dfrac{-3}{r^4}\) QED

Answer 15

So, net force will be repulsive if both dipoles point in the same direction and attractive if the dipoles point in opposite directions.

Answer 16

Cool! That is a much faster method. Thanks for the help!