Induced magnetic field in a cylinder

[Silviu]

Hello! I am confused about the first EM problem on http://web.mit.edu/physics/current/graduate/exams/gen2_F00.pdf (page 4, with the cylinder in a magnetic field). The solution can be found http://web.mit.edu/physics/current/graduate/exams/gen2sol_F00.pdf. In part a the solution is straightforward. However I am a bit confused about part b. They divide the cylinder in solenoids and calculate the field a a given point produced by these solenoids. This makes sense. However, when they plug in the value of $j(t)$ they use the value from part a, and I am not sure why we can do this. In a simple RL circuit you have an ODE that you solve to find the current as a function of time. Here if I understand well, they calculate the induced magnetic field created be $j$ from part a, and use that to calculate the new $j$, but once you take self inductance into account, $j$ from part a is not there anymore. Don't you need to calculate the magnetic field induced by this new $j$ to find $j$ at a later time, and this way you need an ODE? What am I missing here?

 

[Delta2]

You are right if the problem was asking for the exact analytical solution for the current density then you would need to solve a system of partial differential equations (apply Maxwell's equation's in differential form with the boundary conditions imposed by this problem and then solve them, my guess is that the system of PDE's going to be very hard and we can hope only for numerical solutions). (or another method would be to solve an ODE for each solenoid as you said, we would have infinitely many ODE's and they would be coupled due to mutual induction between the solenoids but that's another story).

BUT the problem doesn't ask for the exact analytical solution, it just asks for the next order (of $\omega$) correction to the current density j. In other words it asks for an approximation solution that gives the j up to the first two orders of $\omega$, while from taylor's series we know that j will probably contain more orders, possibly infinite.

 

[Silviu]

You are right if the problem was asking for the exact analytical solution for the current density then you would need to solve a system of partial differential equations (apply Maxwell's equation's in differential form with the boundary conditions imposed by this problem and then solve them, my guess is that the system of PDE's going to be very hard and we can hope only for numerical solutions).

But the problem doesn't ask for the exact analytical solution, it just asks for the next order (of $\omega$) correction to the current density j. In other words it asks for an approximation solution that gives the j up to the first two orders of $\omega$, while from taylor's series we know that j will probably contain more orders, possibly infinite.

Thanks a lot! I feel stupid now for asking this...

 

[Delta2]

Nope don't feel stupid, we all sometimes, just need to read more carefully what the problem asks for :D