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A conducting rod of length l and mass m slides without friction on two parallel rails connected to an initially uncharged capacitor of capacitance C. The whole arrangement is in a uniform magnetic field B. A constant force F is applied to the rod starting at t = 0, with the rod initially at rest. Which statement is correct?

1. The rod moves with constant acceleration
2. The speed of the rod at time t is Ft/(B² l² C + m)
3. The charge on the capacitor varies non-linearly with the rod's velocity
4. The charge on the capacitor never reaches a steady value

Correct answer: The rod moves with constant acceleration

Solution

EMF = B l v, so capacitor charge q = C B l v and current i = C B l a (a = acceleration). Magnetic force opposing motion = B i l = B² l² C a. Newton: F - B² l² C a = m a, giving a = F/(m + B² l² C), a constant. Since a is constant, the rod moves with constant acceleration, and the correct speed expression would be v = F t/(m + C B² l²). Charge q = C B l v is linear in v. The defining correct statement is that acceleration is constant.

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