ATLAS accelerator

Question: In the segment of the ATLAS accelerator shown in section 3.4, suppose the current flowing back and forth between the two cylinders is given by I=a cos bt. What is the charge on one of the cylinders as a function of time?

Solution: We are given the current and want to find the charge, i.e. we are given the derivative and we want to find the original function that would give that derivative. This means we need to integrate:

where q_o is a constant of integration.

We can interpret this in order to explain why a superconductor needs to be used. The constant b must be very large, since the current is supposed to oscillate back and forth millions of times a second. Looking at the final result, we see that if b is a very large number, and q is to be a significant amount of charge, then a must be a very large number as well. If a is numerically large, then the current must be very large, so it would heat the accelerator too much if it was flowing through an ordinary conductor.