well. Unfortunately I have yet to find a fundamental explanation of
superconductivity in metals that works at the introductory level.
Finding charge given current example 8
.In the segment of the ATLAS accelerator shown in figure g, the
current flowing back and forth between the two cylinders is given
byI=acosbt. What is the charge on one of the cylinders as a
function of time?.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:q=∫
Idt=∫
acosbtdt=
a
bsinbt+qo,whereqois a constant of integration.
We can interpret this in order to explain why a superconductor
needs to be used. The constantbmust 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 ifbis a very large
number, andqis to be a significant amount of charge, thenamust
be a very large number as well. Ifais 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.Constant potential throughout a conductor
The idea of a superconductor leads us to the question of how
we should expect an object to behave if it is made of a very good
conductor. Superconductors are an extreme case, but often a metal
wire can be thought of as a perfect conductor, for example if the
parts of the circuit other than the wire are made of much less con-
ductive materials. What happens ifRequals zero in the equation
R= ∆V/I? The result of dividing two numbers can only be zero if
the number on top equals zero. This tells us that if we pick any two
points in a perfect conductor, the voltage difference between them
must be zero. In other words, the entire conductor must be at the
same potential.542 Chapter 9 Circuits