How Does a Proton Behave in a Uniform Magnetic Field?

[Moneer81]

Homework Statement

A proton is placed in a uniform 1.5T magnetic field.
a. Quantify the magnitude (algebraically) of the force and torque on the proton.
b. Describe the time evolution of the proton's magnetic moment.
c. Describe how to generate a uniform 1.5T magnetic field from a 1.0 amp source over a volume of 20cm x 20cm x 40cm using coils of wire.

Homework Equations

The Attempt at a Solution

a. So I am assuming the proton's magnetic moment is pointing to the right, and the external magnetic field is pointing up. Then the torque on the proton will be given by:

torque = mu x B = (14.1x10^-23 J/T)(1.5T)(sin 90) = 21.15x10^-23 J

force = qV x B. Can I find the force? I don't know anything about the proton's velocity. I am just assuming it is precessing around an axis that points to the right of the page. Am I on the right track here?

b. After some time, the magnetic moment of the proton (that was initially pointing to the right of the page) will align with the magnetic field B. Since there are two different spin states, the magnetic moment can be parallel with the magnetic field (low energy) or anti-parallel (high energy). DOes that sound right?

c. OK so is there an easy way to work with a volume like that? Isn't easier to create a magnetic field in a coil or cylinder? I know that we have wrap the wire coil around the volume but what else can we say to answer that question?

Thanks for taking the time to read my post.

 

[anathema]

a. Your calculations for the torque and force on the proton seem to be correct. However, remember that the force on a charged particle in a magnetic field is given by F = qv x B, where v is the velocity of the particle. In this case, since the proton is not moving, the force would be 0.

b. Your understanding of the time evolution of the proton's magnetic moment is correct. As the proton precesses in the magnetic field, it will eventually align itself with the field, resulting in a lower energy state. This is known as Larmor precession.

c. To generate a uniform 1.5T magnetic field over a volume of 20cm x 20cm x 40cm, you can use a solenoid coil with a current of 1.0 amp. The magnetic field inside a solenoid is given by B = u0nI, where u0 is the permeability of free space, n is the number of turns per unit length, and I is the current. By adjusting the number of turns and the length of the solenoid, you can create a uniform magnetic field over the desired volume. You can also use multiple solenoids placed next to each other to create a larger uniform field.

 

[Moetasim]

a. Your calculations for the torque and force seem to be correct. However, keep in mind that the force on a charged particle in a magnetic field is given by F = qv x B, where v is the velocity of the particle. So if we assume the proton is moving with a velocity v, then we can find the force on the proton.

b. Yes, you are correct in saying that the magnetic moment of the proton will align with the magnetic field after some time. This is known as precession, and it is a result of the proton's spin interacting with the external magnetic field. The proton's magnetic moment will continue to precess around the axis of the external magnetic field as long as it is in that field.

c. One way to generate a uniform magnetic field is by using a Helmholtz coil setup. This involves two identical coils placed parallel to each other with a distance equal to the radius of the coils. The current through each coil should be in the same direction, and the magnitude of the current should be such that it produces a magnetic field of 1.5T at the center of the coils. By adjusting the dimensions and current of the coils, a uniform magnetic field can be created over a specific volume.