Charged fractal in a magnetic field

Electricity and Magnetism Level 4

We generate our fractal as follows:

A stick of total length \(L\) is taken. A certain amount of charge is uniformly distributed along it's length.

Now, one third of the stick is removed from the middle. We are now left with two sticks which are one third the size of the original stick. Remove one third from the middle, from these two sticks...and from the smaller sticks left, ad infinitum.

What we are left with looks like the figure. Now, after these infinite iterations, let the fractal that we get have a mass \(M\) and a total charge \(Q\) residing on it.

The fractal is spun with an angular velocity \(\omega\) about the center of the original stick.

A magnetic field of intensity \(B\) exists parallel to the axis of rotation.

Let the magnitude of rotational kinetic energy associated with the fractal be \(A\) and the magnitude of it's magnetic potential energy be \(B\).

Then find \(A+B\) in joules.

Details and Assumptions

  • \(M=12 \text{ kg}, Q=6 C, B = 1 \text{ T}, \omega = 9 \text{ rad/s}, L = 7 \text {m} \)

  • Assume the standard convention for magnetic potential energy (meaning, for what configuration the P.E is taken to be 0).

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