EDIT ** : I've found the solution but cannot understand it for the life of me, can someone just explain what they are doing? Even the first step, I don't understand why they get a 2pi * 1 in the numerator, I know that H_phi = I / 2pi * rho, and then B = Mu_0 * H, but where the heck is that extra 2pi coming from? Then I have no idea why they are using current sheet density K...Please explain. Thanks!
A current of -100 a_z A/m flows on the conducting cylinder of radius 5mm, and 500 a_z A/m is present on the conducting cylinder of radius 1mm. Find the magnitude of the force per meter length that is acting to split the outer cylinder apart along its length.
I'm really not sure how to go about this. I'm starting by saying that the magnetic flux density inside a cylindrical conductor is:
B_phi = I * (rho) / (2 * pi * radius^2)
So if I call the inner cylinder radius r_1 and the outer r_2, and inner current I_1 and outer I_2, then for rho < 1mm:
B_phi = rho [ I_1/ (2pi * r_1^2) + I_2 / (2pi * r_2^2) ]
And I know that the current density outside of the inner cylinder is B_phi = I / 2pi * rho, so for 1mm < rho < 5mm I get:
B_phi = (rho * I_2) / (2pi * r_2^2) + I_1 / (2pi * rho)
But do I also need to consider the region outside 5mm? Or since I'm considering the force at r = 5mm, can I just stop here?
Which leads me to my next question, the force on a line is:
F = -I int[ B cross DL]
Where in this case DL = rho d(rho), and clearly this force must be in the a_z direction (as required).
But, what current do I use for I? or do I integrate each separately, meaning do I integrate the part from 0 to 1mm and then 1mm to 5mm as two different integrals?
The answer in the book is: 4pi * 10^(-5) N/m
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