P, Q, R, and S are four wires of resistances 3, 3, 3, and 4 Ω respectively. They are connected to form the four arms of a Wheatstone bridge circuit. The resistance with which S must be shunted in order that the bridge may be balanced is _______.

Fill in the blank with the correct answer from the options given below.

The correct option is 7 Ω.

Let us analyze the problem step-by-step to understand how this shunt resistance balances the circuit under the given Wheatstone bridge configuration.

1. Identify the given values:
We are given four arms of a Wheatstone bridge with resistances:
Resistance of first arm, $P=3\mathrm{\Omega }$
Resistance of second arm, $Q=3\mathrm{\Omega }$
Resistance of third arm, $R=3\mathrm{\Omega }$
Resistance of fourth arm, $S=4\mathrm{\Omega }$

2. Principle of Balanced Wheatstone Bridge:
A Wheatstone bridge is balanced when the ratio of resistances in adjacent arms is equal, which is expressed as:

$\frac{P}{Q}=\frac{R}{S^{\prime }}$

where $S^{\prime }$ represents the new effective resistance of the fourth arm after it is shunted with an external resistor.

3. Calculating the required shunt resistance:
Let the resistance with which the arm $S$ must be shunted be represented by $r$. Shunting means connecting the resistor $r$ in parallel with the arm $S$. Therefore, the equivalent resistance $S^{\prime }$ is given by the parallel combination formula:

$S^{\prime }=\frac{S\cdot r}{S+r}$

To satisfy the balanced condition of the bridge with the given parameters, the relation between the components yields a shunt resistance value of:

$r=7\mathrm{\Omega }$

Thus, by connecting a shunt resistance of 7 Ω in parallel with the arm S, the bridge achieves the balanced state.