The acceleration of protons using a changing magnetic field

[Jaccobtw]

Homework Statement

In earlier learning sequences we described how a static magnetic field cannot change the speed (and therefore kinetic energy) of a free charged particle. A changing magnetic field can, and this is one way particle beams are accelerated. Consider free protons following a circular path in a uniform magnetic field with a radius of 1m. At t=0s, the magnitude of the uniform magnetic field begins to increase at 0.001T/s. Enter the tangential acceleration of the protons in m/s^2: positive if they speed up and negative if they slow down.

Relevant Equations

$$F = qvBsin\theta$$
$$ r = \frac{mv}{qB}$$
$$\Phi = \int B \cdot dA$$

If we increase the magnetic field, the radius of the particle's circular path will decrease which increases the tangential acceleration. How do I find the tangential acceleration. Do I use derivatives?

 

[robphy]

Note that since ##\vec F_{mag} = q\vec v \times \vec B##, we necessarily have ##\vec F_{mag} \perp \vec v##. So, ##\vec F_{mag}## can't change the magnitude of ##\vec v##.

Are there other relevant equations needed?
A labeled diagram would help.

 

[Delta2]

A time varying magnetic field generates an electric field which will give the tangential acceleration to the protons. Use Faraday's law to calculate the magnitude of the generated electric field at r=1m.
The field lines of this electric field are concentric circles, centered at the origin, with their plane perpendicular to the uniform magnetic field.

 

[Jaccobtw]

A time varying magnetic field generates an electric field which will give the tangential acceleration to the protons. Use Faraday's law to calculate the magnitude of the generated electric field at r=1m.
The field lines of this electric field are concentric circles, centered at the origin, with their plane perpendicular to the uniform magnetic field.

Thanks for the big hint!. The protons are accelerated by the electric field.

 

[robphy]

Thanks for the big hint!. The protons are accelerated by the electric field.

My hint was a little more subtle.
Note that, when the magnetic force is the only force,
the protons are accelerated by the magnetic field
since their velocity vectors are turned by the magnetic force.

 

[Jaccobtw]

My hint was a little more subtle.
Note that, when the magnetic force is the only force,
the protons are accelerated by the magnetic field
since their velocity vectors are turned by the magnetic force.

But doesn't the increasing tangential velocity only come from the electric field? The magnetic force is perpendicular to the proton's motion so it's speed can't increase from that, only direction can change.

 

[robphy]

But doesn't the increasing tangential velocity only come from the electric field? The magnetic force is perpendicular to the proton's motion so it's speed can't increase from that, only direction can change.

Yes, the speed changes because of the electric field,
but
##\vec a=\frac{d\vec v}{dt}## is a nonzero vector
when the magnitude of ##\vec v## changes, or the direction of ##\vec v## changes, or both.

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