# Force & Torque on Electric Dipole in Magnetic Field

• Snarlie
In summary: The torque on the electric dipole is given by\vec{M}=\vec{r} \times \vec{F}=(\vec{r} \cdot \vec{p_e}\cdot \vec{r})(-\frac{\mu_0}{4\pi}\cdot\frac{\vec{p_m}}{r^3})If you want to calculate the torque, you need to solve for ##\vec M##.
Snarlie
Homework Statement
Equation for an electric dipole in a magnetic field
Relevant Equations
none
Snarlie said:
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}
Would this also be correct? Or is there something wrong with it?

Snarlie

Last edited:
Delta2
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 ?
Snarlie said:
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.

Last edited:
BvU said:
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?

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.

## What is a force and torque on an electric dipole in a magnetic field?

A force and torque on an electric dipole in a magnetic field describes the effect of a magnetic field on an electric dipole, which is a pair of equal and opposite charges separated by a distance. When a dipole is placed in a magnetic field, it experiences a force and torque due to the interaction between the magnetic field and the dipole's charges.

## What is the formula for calculating the force on an electric dipole in a magnetic field?

The formula for calculating the force on an electric dipole in a magnetic field is F = qv x B, where F is the force, q is the charge of the dipole, v is the velocity of the dipole, and B is the strength of the magnetic field. This formula is known as the Lorentz force law.

## What is the direction of the force on an electric dipole in a magnetic field?

The direction of the force on an electric dipole in a magnetic field is perpendicular to both the magnetic field and the velocity of the dipole. This means that the force will cause the dipole to rotate, with one end moving towards the magnetic field and the other end moving away.

## What is the formula for calculating the torque on an electric dipole in a magnetic field?

The formula for calculating the torque on an electric dipole in a magnetic field is T = p x B, where T is the torque, p is the dipole moment (the product of the charge and the distance between the charges), and B is the strength of the magnetic field. This formula is derived from the cross product of the force and the distance between the charges.

## What factors affect the force and torque on an electric dipole in a magnetic field?

The force and torque on an electric dipole in a magnetic field are affected by the strength of the magnetic field, the charge and separation of the dipole's charges, and the velocity of the dipole. The angle between the magnetic field and the dipole's velocity also affects the force and torque. Additionally, the size and shape of the dipole can also impact the force and torque it experiences in a magnetic field.

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