In the circuit shown, the resistance r is a variable resistance. If for r = f R , the heat generation in r is maximum then the value of f is

Step-by-step solution

Power in “r” P = I 2 2 r = [ I × R R + r ] 2 r = I 2 R 2   r ( R + r ) 2 P = [ E R + R × f R R + f R ] 2 R 2 ( f R ) ( R + f R ) 2 = E 2 R 2 f R R 2 [ 1 + f ( f + 1 ) ] 2 R 2 ( 1 + f ) 2 P = E 2 f R ( 1 + 2 f ) 2 For P ( max )   d P d f = 0 E 2 R = [ ( 1 + 2 f ) 2 − f 2 ( 1 + 2 f ) ( 0 + 2 ) ( 1 + 2 f ) 2 ] = 0 ( 1 + 2 f ) 2 = 4 f ( 1 + 2 f ) 1 + 2 f = 4 f f = 1 2