Force & Torque on Electric Dipole in Magnetic Field

[Snarlie]

Homework Statement

Equation for an electric dipole in a magnetic field

Relevant Equations

none

Homework Statement: An electric dipole is in an equatorial field of a magnetic dipole. What force and torque does the electric dipole feel for its different orientations and different directions of movement.
Homework Equations: Many, written below.

\usepackage[utf8]{inputenc}Hi, I need help for this homework. Can you review the following process that I've done and tell me if it is okay or what it is wrong with it.

I'm going to be using a few of different identities where a is a constant vector.
\begin{align}
\nabla f(r) &= \frac{df}{dr}\nabla r \\
\nabla r &= \frac{\vec{r}}{r} \\
\nabla \vec{a} &= \vec{0} \\
(\vec{u} \cdot \nabla)(f\vec{v}) &= \vec{v}(\vec{u} \cdot \nabla f) + f(\vec{u} \cdot \nabla)\vec{v} \\
\vec{u} \times (\vec{v} \times \vec{w}) &= (\vec{u} \cdot \vec{w})\vec{v} - (\vec{u} \cdot \vec{v})\vec{w}
\end{align}
Which I will refrence in the upcoming calculations.

I've attached a file of a sketch.

Lets begin with a magnetic field of a magnetic dipole in its equatorial plane:
\begin{align}
\vec{B}(\vec{r}) &= \frac{\mu_0}{4\pi}\cdot\frac{3\vec{r}(\vec{p_m}\cdot\vec{r})-\vec{p_m}r^2}{r^5} \\
\vec{p_m}\perp\vec{r} &\Rightarrow \vec{p_m}\cdot\vec{r}=0 \\
\vec{B}(r)&= -\frac{\mu_0}{4\pi}\cdot\frac{\vec{p_m}}{r^3} \\
\end{align}The force that an electric dipole feels in a magnetic field is:
\begin{align}
\vec{F} &= \vec{v}\times(\vec{p_e}\cdot\nabla)\vec{B}(r) \\
\vec{F} &= \vec{v}\times(\vec{p_e}\cdot\nabla)(-\frac{\mu_0}{4\pi}\cdot\frac{\vec{p_m}}{r^3}) \\
\vec{F} &= -\frac{\mu_0}{4\pi}\vec{v}\times(\vec{p_e}\cdot\nabla)(\frac{\vec{p_m}}{r^3}) \\
\vec{F} &\stackrel{(4)}{=} -\frac{\mu_0}{4\pi}\vec{v}\times(\vec{p_m}(\vec{p_e}\cdot\nabla(\frac{1}{r^3}))+\frac{1}{r^3}(\vec{p_e}\cdot\nabla)\vec{p_m}) \\
\vec{F} &\stackrel{(1),(2),(3)}{=} -\frac{\mu_0}{4\pi}\vec{v}\times(\frac{-3}{r^5}\vec{p_m}(\vec{p_e}\cdot\vec{r}))
\end{align}
So basically, I have a few problems with the end result. Firstly, I assumed that electric dipole and magnetic dipole are constant vectors. Is this correct? Also, was the inital equation for force on an electric dipole in a magnetic field correct?
My end result tells me that the force on electric dipole is non existent if
\begin{align}
\vec{p_e}\perp\vec{r} &\Rightarrow \vec{p_e}\cdot\vec{r}=0 \\
\end{align}
and in that case also non-dependent on the movement of electric dipole. I find this hard to believe, can someone explain what I did wrong and how to fix it?

I've also calculated the torque:
\begin{align}
\vec{M} &= \vec{r} \times \vec{F} \\
\vec{M} &= \frac{3\mu_0}{4\pi r^5}\vec{r} \times \vec{v} \times (\vec{p_m}(\vec{p_e}\cdot\vec{r})) \\
\vec{M} &\stackrel{(5)}{=} \frac{3\mu_0}{4\pi r^5} ((\vec{r}\cdot\vec{p_m})(\vec{p_e}\cdot\vec{r}))\vec{v} - (\vec{r}\cdot\vec{v})(\vec{p_e}\cdot\vec{r})\vec{p_m})
\end{align}

Would this also be correct? Or is there something wrong with it?

I appreciate all your responses!

Snarlie

 

[BvU]

Hello Snarlie, !

You want to read the PF guidelines -- 'dunno' isn't good enough in this forum !

An electric dipole can be thought of as two opposite charges, separated by a small distance d. What is the force exerted by a magnetic field on an electric charge ?

my homework

Complete problem statement ?

 

[Snarlie]

Thx for the reply, I'll recreate the post when I have the time to write up all the things I've done.

 

[Snarlie]

Hello Snarlie, !

You want to read the PF guidelines -- 'dunno' isn't good enough in this forum !

An electric dipole can be thought of as two opposite charges, separated by a small distance d. What is the force exerted by a magnetic field on an electric charge ?
Complete problem statement ?

I've re-edited the inital problem, so it has a lot more explanation and I also showcased what I have done. I'am woried though, that since many people saw the first post which had no explanation, they won't give this thread another look. Should I repost the problem in hopes of getting people's attention?

 

[TSny]

Hello, Snarlie. I think your starting expression for the force, equation (9), is correct for non-relativistic speed of the electric dipole.

The force on the electric dipole involves evaluating spatial derivatives of $\vec B$ and then evaluating in the equatorial plane of the magnetic dipole. So, you have to wait until after you take the derivatives before restricting to the equatorial plane. This looks a little messy. I have not tried to do the calculation.