In chapter 5, magnetostatics, of Griffiths' Introduction to Electrodynamics (third edition), there's a problem in the back of the chapter that asks you to calculate the force of attraction between the northern and southern hemispheres of a spinning charged spherical shell.
The problem in its exact words:
Problem 5.42: Calculate the magnetic force of attraction between the northern and southern hemispheres of a spinning charged spherical shell.
Well, we haven't gotten so far in our class that I can actually do this. But I'm reviewing for the test, so I figured there was at least some part of the problem that I could do. I decided to calculate the magnetic force at the southern tip P. Seeing as this isn't the problem, I don't know if my answer is correct. Here's what I did:
I used the Biot-Savart law to do so. I let the distance from P to the point of current dI be r, the height (which goes from 0 to 2R) be z, and the distance from z to dI be a. Radius = R.
I used these values to work out a formula for r, namely r^2= z^2 + (2Rz)^2. My current I = σω(2Rz - z^2)^(1/2) (though I'm thinking it should be line charge λ). The diameter is equal to 2∏(2Rz - z^2)^(1/2)
My Biot savart law has me integrate I/r^2 cos∅ which turns into:
Resulting in the magnetic field B equal to μσω/3 (which seems unlikely to me.)
If formatting this or scanning my work to this post would motivate someone to actually verify my answer, then I'll do that... just let me know.
The problem in its exact words:
Problem 5.42: Calculate the magnetic force of attraction between the northern and southern hemispheres of a spinning charged spherical shell.
Well, we haven't gotten so far in our class that I can actually do this. But I'm reviewing for the test, so I figured there was at least some part of the problem that I could do. I decided to calculate the magnetic force at the southern tip P. Seeing as this isn't the problem, I don't know if my answer is correct. Here's what I did:
I used the Biot-Savart law to do so. I let the distance from P to the point of current dI be r, the height (which goes from 0 to 2R) be z, and the distance from z to dI be a. Radius = R.
I used these values to work out a formula for r, namely r^2= z^2 + (2Rz)^2. My current I = σω(2Rz - z^2)^(1/2) (though I'm thinking it should be line charge λ). The diameter is equal to 2∏(2Rz - z^2)^(1/2)
My Biot savart law has me integrate I/r^2 cos∅ which turns into:
B = σμω/2 * Integral[(z*diameter)/(r^(3/2))]
Resulting in the magnetic field B equal to μσω/3 (which seems unlikely to me.)
If formatting this or scanning my work to this post would motivate someone to actually verify my answer, then I'll do that... just let me know.
