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Constant Magnetic Field, Magnetics

Problem 3.265

A parallel-plate capacitor with area of each plate equal to \(S\) and the separation between them to \(d\) is put into a stream of conducting liquid with resistivity \(\rho .\) The liquid moves parallel to the plates with a constant velocity \(v .\) The whole system is located in a uniform magnetic field of induction \(B\), vector \(\mathbf{B}\) being parallel to the plates and perpendicular to the stream direction. The capacitor plates are interconnected by means of an external resistance \(R\). What amount of power is generated in that resistance? At what value of \(R\) is the generated power the highest? What is this highest power eaual to?

Reveal Answer
 3.265. P=v2B2d2R/(R+ρd/S)2; when R=ρd/S, the power  is P=Pmax⁡=1/4v2B2dS/ρ\begin{aligned} &\text { 3.265. } P=v^{2} B^{2} d^{2} R /(R+\rho d / S)^{2} ; \text { when } R=\rho d / S, \text { the power }\\ &\text { is } P=P_{\max }=1 / 4 v^{2} B^{2} d S / \rho \end{aligned}
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