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 respectivity `rho`. The liquid moves parallel to the plates with a constant velocity `v`. The whoel system is located in a unifrom magentic field of induction `B`, vector `B` being parallel to the plates are interconnected by means of an exteranal resistance `R`. What amount of power is genrated in that resistance? At what value of `R` is the generated power the highest? What is this highest power equla to ?

Explanation

Resistance of the liquid between the plates `= (rho d)/(S)`
Voltage beween the plates `= Ed = v Bd`,
Current through the plates `= (vBd)/(R + (rho d)/(S))`
Power generted, in the external resistance `R`,
`P = (v^(2) B^(2) d^(2) R)/((R + (rho d)/(S))^(2)) = (v^(2) B^(2) d^(2))/((sqrt(R) + (rho d)/(S sqrt(R)))^(2)) = (v^(2) B^(2) d^(2))/([{R^(1//4) - ((rho)/(S sqrt(R)))^(1//2) }^(2) + 2 sqrt((rho d)/(S))]^(2))`
This is maximum when `R = (rho D)/(S)` and `P_(max) = (v^(2) B^(2) Sd)/(4 rho )`