# Particle/dipole acceleration in non-uniform magnetic field

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1. Nov 21, 2013

### kmm

I've come across a few places that mention that dipoles and charged particles accelerate in non-uniform magnetic fields. Is this true? If the Magnetic force is always perpendicular to the velocity of a charge, I don't see why it would accelerate. I see it having centripetal acceleration with constant kinetic energy unless the centripetal acceleration is all that's needed for the charge to emit a photon. I also don't see a dipole would accelerate in a non-uniform magnetic field.

2. Nov 21, 2013

### WannabeNewton

A charged particle with some non-zero velocity will accelerate in any magnetic field as per the Lorentz force law, given that the velocity and magnetic field are not parallel. Simple as that.

As for magnetic dipoles, if we have an infinitesimal current loop with magnetic dipole moment $\vec{m}$ in a magnetic field $\vec{B}$ then it can be shown that it experiences a force $\vec{F} = \vec{\nabla}(\vec{m}\cdot \vec{B})$. See problem 6.4 in Griffiths.

3. Nov 21, 2013

### kmm

Except the Lorentz force law says that the magnetic force is $\mathbf{F_{magnetic}} = q (\mathbf{v} \times \mathbf{B})$ so the force is perpendicular and therefore no acceleration, unless you're simply referring to centripetal acceleration.

I will check Griffiths. Also, what then is the advantage of a non-uniform magnetic field in particle accelerators or like that used in the Stern-Gerlach experiment? Simply to control the path better?

4. Nov 21, 2013

### WannabeNewton

The force being perpendicular to the velocity doesn't imply that there is no acceleration. A force always imparts an acceleration as per Newton's 2nd law. What makes you think centripetal acceleration is not an acceleration?

Well in the Stern-Gerlach experiment we need a non-uniform magnetic field in order for an interacting magnetic dipole to get deflected in a way that distinguishes between different dipole moments. Classically this goes back to that equation I wrote above: $\vec{F} = \vec{\nabla}(\vec{m}\cdot \vec{B})$ specialized to a constant dipole moment. See section 1.1 of Sakurai's QM text.

5. Nov 21, 2013

### kmm

I don't think my original question was asked well and I did imply that centripetal acceleration isn't acceleration. What I meant was that I wasn't sure if it was the magnitude of the velocity that changed since that's the most common context I've heard it in. OK, so yes there is an acceleration, just no change in speed.

Unfortunately, I don't have Sakurai. Although, if the force on the dipole is $\vec{F} = \vec{\nabla}(\vec{m}\cdot \vec{B})$ then it seems the dipole would be deflected whether the magnetic field was uniform or not.

6. Nov 21, 2013

### WannabeNewton

If the magnetic field is uniform then the gradient will vanish and there won't be a force.

7. Nov 21, 2013

### kmm

Oh of course, thanks for helping me clear that up.