# Homopolar generator

I have proven the induced emf between the centre and rim in a circular disc of radius a and angular velocity w, with a magnetic field B parallel to its axis is 0.5wa2B.

I need to find the time taken for the disc to slow to half it’s initial speed ignoring friction, given a resistance R is connected between centre and rim and all other circuit resistance is negligible. It has mass m.

So I know it’s moment of inertia is 0.5ma2, and that energy is conserved such that the difference in the rotational KE initially and finally equals the energy dissipated in the resistance. This energy E is
E=0.5Iw2-0.5I(0.5w)2
E=0.5Iw2-0.125Iw2
E=(3/8)(0.5ma2)w2
E=(3/16)ma2w2

Now comes the problem in calculating the energy dissipated in the resistance. Obviously the induced emf and so induced current are time dependent. I’m not sure how I can get expressions for these in terms of time such that I could get the power dissipation with time and integrate for the energy. Any clues please? Thanks 🙂

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