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 basicly, 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}
!