In the above question, show that power output of electri motor is maximum when the back emf is one-half of the sourece emf, provided the resistance of the winding of the motor is negligible .

Explanation

Here, emf of d.c. source `= epsilon`,
resistance of motor R = 0
Internal resistance of source = r
Let `epsilon'` be the back emf of motor. Therefore current in the circuit, `I = (epsilon - epsilon')/ (r + R) = (epsilon - epsilon')/r`
Power output of the motor,
P = power output of source ` = epsilon I - I^(2) r`
P is maximum when `(dP)/(dI) =0`
or `(d)/(dt) (epsilon I - I^(2)r) = 0 or epsilon = 2 I r`
or `I = epsilon/(2r) or (epsilon - epsilon')/r = epsilon/(2r) or epsilon' = epsilon/2`