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Faraday proposed generating current from a flowing river: two identical conducting plates, each of length a and width b, are placed parallel and facing each other on opposite banks, a distance d apart, with the river flowing between them at velocity u. The plates are joined by a load resistance R. Taking the vertical component of Earth's magnetic field as Bv and the resistivity of river water as rho, the current through the load R is:

1. Bv*u*b / R
2. Bv*u*b / (R + rho*d/(a*b))
3. Bv*u*d / (R + rho*d/(a*b))
4. None of the above

Correct answer: Bv*u*b / (R + rho*d/(a*b))

Solution

The moving conducting water acts as a rod of width (plate separation). The motional EMF across the plates is e = Bv * u * b, where b is the dimension perpendicular to both flow and field appropriate to the plate width. The internal resistance is that of the water column between the plates: r = rho * d/(A) = rho*d/(a*b), where A = a*b is the plate face area and d the separation. Current I = e/(R + r) = Bv*u*b / (R + rho*d/(a*b)).

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