The liquid is conducting and hence contains free electrons. The electrons in the field \\vec{B} experiences force Bev . This force is in y-direction and the electrons drift in this direction and this builds up a potential between the plates in such a way that \\dfrac{V}{d} = Bev The resistance of the liquid between the plates is \\dfrac{\\rho d }{S} The total resistance is \\dfrac{\\rho d }{S} + R The current in the external resistance is I = \\dfrac{V}{ R + \\dfrac{\\rho d }{S} }=\\dfrac{Bvd}{R + \\dfrac{\\rho d }{S}} Power P = I^2R = \\dfrac{V}{ R + \\dfrac{\\rho d }{S} }=(\\dfrac{Bvd}{R + \\dfrac{\\rho d }{S}})^2R