Answer to Question #112525 in Electricity and Magnetism for Joel

Question #112525
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?
1
Expert's answer
2020-04-28T09:46:39-0400

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:


E=ΔΦΔt=iR, iΔt=q, ΔΦΔt=qΔtR. q=ΔΦR.\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


ΔΦ=BA=μ0I2RaπRa2=πμ0IRa2.\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=πμ0IRa2R.q=\frac{\pi\mu_0IR_a}{2R}.

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