A single circular loop of wire with initial radius r0=0.040m is mounted on a shaft so that the loop rotates about its diameter. The shaft spins with constant angular speed ω=120rads−1 in a uniform magnetic field of magnitude B0=0.18T directed horizontally.
At the same time, a motor slowly increases the loop’s radius according to r(t)=r0(1+αt) with α=2.0×10−3s−1, while the field magnitude is gradually reduced by external coils according to B(t)=B0(1−βt) with β=5.0×10−4s−1. All functions are valid for times t≪1/α and t≪1/β so that the loop remains intact and the field remains positive.
(a) Derive an expression for the magnetic flux through the loop as a function of time for small t.
(b) Obtain an explicit expression for the induced emf E(t) for small t.
(c) Determine the first time t>0 at which the induced emf is zero.
(d) State physically why the emf vanishes at that instant despite all three quantities changing.