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Find the current in the sliding rod AB (...

Find the current in the sliding rod AB (resistance = R) for the arrangement shown in Fig. B is constant and is out of the paper. Parallel wires have no resistance. `upsilon` is constant. Switch is closed at time t = 0.

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In Fig. magnetic field `vec B` is constant and is out of paper. When the sliding rod AB moves with a velocity `upsilon` in the direction shown, induced current in AB is from A to B. As switch S is closed at time t = 0, the capacitor gets charged. If Q (t) is charge on the capacitor, then
`I = (dQ)/(dt) = (B d upsilon)/(R ) - (Q)/(RC)` or `(Q)/(RC) + (dQ)/(dt) = (B d upsilon)/(R )`
`Q = upsilon` A is an arbitrary constant. ltbRgt At `t = 0, Q = 0`
`:. A = - upsilon B d C`
Put in (i), `Q = upsilon B d C [1 - e^(-t//RC)]`
Differentiating w.r.t. t, we get `I = (dQ)/(dt) = upsilon B d C xx (1)/(RC) e^(-t//RC) = (upsilon B d)/(R ) e^(-t//RC)`
Find is the current in the sliding rod.
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