Answer to Question #112525 in Electricity and Magnetism for Joel

Question

Two concentric conducting loops have radii Ra and Rb (Ra>>Rb). The inner loop has
current I flowing counter-clockwise through it, and the outer loop has no current. The outer
loop has resistance R. The inner loop is cut by scissors, so that in a fairly small duration
current in it drops to zero. What is the direction of induced current in the outer loop? How
much total charge passes a point in the loop during the period this current flows?

Answer 1

According to Lenz's law, the current in the outer loop will try to generate a magnetic field that would prevent the field in the inner loop to cease. Therefore, according to the figure below and the right-hand rule, the current in the outer loop will flow counter-clockwise as well.

According to Faraday's and Ohm's law:

$$\Epsilon=\frac{\Delta \Phi}{\Delta t}=iR,\\ \space\\ i\Delta t=q,\\ \space\\ \frac{\Delta \Phi}{\Delta t}=\frac{q}{\Delta t}R.\\ \space\\ q=\frac{\Delta \Phi}{R}.$$

The inner loop creates the flux that disappears of

$$\Delta\Phi=BA=\frac{\mu_0I}{2R_a}\pi R_a^2=\frac{\pi\mu_0IR_a}{2}.$$

Substitute this into the expression for the charge:

$$q=\frac{\pi\mu_0IR_a}{2R}.$$