Four resistors are connected to form a Wheatstone bridge ABCD . The resistances of the arms are R_ AB = R , R_ BC = x , R_ AD = 4 , and R_ CD = 12 . A galvanometer connected between B and D shows no deflection. An ideal battery of 24 V is connected across A and C . If the total electrical power dissipated in the circuit is 48 W , the value of the unknown resistance x is :

Step-by-step solution

Since the galvanometer shows no deflection, the Wheatstone bridge is balanced. The condition for a balanced bridge is: R_ AB R_ BC = R_ AD R_ CD R x = 4 12 = 1 3 x = 3R The total power dissipated in the circuit is given by P = V^2 R_ eq , where R_ eq is the equivalent resistance of the circuit. 48 = (24)^2 R_ eq R_ eq = 576 48 = 12 The circuit consists of two parallel branches: branch ABC with resistance R_ ABC = R + x = 4R , and branch ADC with resistance R_ ADC = 4 + 12 = 16 . The equivalent resistance is: 1 R_ e