What is the Required Distance for 25% Transmission in a Spin 1/2 Particle Beam?

[nmsurobert]

Homework Statement

A beam of spin 1/2 particles with speed v passes through a series of SGz devices. The first SGz device transmits particles with Sz = ħ/2 and filters out particles with Sz=-ħ/2. The second SGz device transmits particles with Sz=-ħ/2 and filters out particles of Sz=ħ/2. Between the two devices is a region of length l in which there is a uniform magnetic field B pointing in the x-direction. Determine the smallest value of l such that exactly 25% of the particles transmitted by the first SGz Device are transmitted by the second device. Express your result in terms of ω=egB/2mc and v.

Homework Equations

The Attempt at a Solution

Because of the way particles are being passed through the two SGz devices doesn't only 25% of the particles pass through the second device regardless of the distance between the SGz devices?
I know that's not exactly right but I'm failing understand why it's not right.
I've been pulling my hair out. I have no idea where this 'l' variable is coming from.
Any advice or direction would be appreciated.

 

[TSny]

Make sure you're not thinking of the B field in the x direction as acting like a SGx device.

 

[nmsurobert]

I don't think I'm thinking of it as an SGx device. So the magnetic field just changes the spin over time, right? But that's for B in the z-direction. Does B in the X-direction do the same thing?

 

[TSny]

So the magnetic field just changes the spin over time, right? But that's for B in the z-direction. Does B in the X-direction do the same thing?

As the particle passes through the B_x field, the field alters the spin state of the particle. You need to know specifically what the B_x field does to the spin state. The net change in the spin state due to the B_x field depends on how much time the particle spends in the field. That's why the distance L matters.

 

[nmsurobert]

Ok. I understand why it's important. Butbim having trouble finding an 'l' anywhere in my book or notes. I don't know where the variable is supposed to be derived from.
Is 'l' an angle? That's all I can find in relation to the equations given at the end of the problem.

 

[nmsurobert]

Thank you for your help thus far btw.

 

[TSny]

Ok. I understand why it's important. Butbim having trouble finding an 'l' anywhere in my book or notes. I don't know where the variable is supposed to be derived from.
Is 'l' an angle? That's all I can find in relation to the equations given at the end of the problem.

In this font, lower case L looks too much like upper case I. Is it OK if we use 'L' instead of 'l' for the distance the particle travels through the uniform field B_x? L is not an angle, it's a distance. Can you relate the time, t, that the particle spends in the region of B_x to the distance L?

 

[nmsurobert]

Well if it's moving at speed v and v = change in distance/time. And L is the distance we're trying to measure. There's a relation I suppose.

 

[TSny]

As preparation for this problem, you should have studied what happens to the spin of a particle when placed in a uniform magnetic field.