Question

In a Wheatstone’s bridge all the four arms have equal resistance R. If the resistance of the galvanometer is also R, the equivalent resistance of the combination across the battery is:
(A)R/2
(B)R
(C) 2R
(D)R/4

Answer 1

A Wheatstone Bridge is an electrical circuit used to measure an unknown resistance when the balanced condition of the two legs of the bridge circuit is satisfied. When the balanced condition is satisfied, no current is measured across the galvanometer of the circuit which is used to measure the electrical current across the 2 arms.

Complete answer:
A Wheatstone bridge circuit is given as follows:

Fig: circuit diagram of wheatstone bridge where A, B, C, D are the resistances and G is the galvanometer.
$\because$ All the resistances in the wheatstone bridge are R, the balanced condition of wheatstone bridge is satisfied and as a result, no current flows through the galvanometer coil. Thus, the galvanometer from the circuit can be neglected while calculating the equivalent resistance in the circuit.
Thus the circuit becomes equivalent to the circuit below in the absence of the galvanometer.

Each of the resistors has a resistance of R. Thus, in the above circuit, A and C are in series and B and D are in series. The equivalent value both these series combinations is obtained as 2ohm each and both are parallel to one another and thus we calculate the equivalent resistance for parallel combination as follows:
$\begin{aligned} & \dfrac{1}{{{R}_{eq}}}=\dfrac{1}{2R}+\dfrac{1}{2R} \\\ & \Rightarrow {{R}_{eq}}=R\Omega \\\ \end{aligned}$
Thus, the equivalent resistance of the combination across the battery is calculated as R.

So, the correct answer is “Option B”.

Additional Information:
The balanced condition of the Wheatstone Bridge is given as follows in relation to the figure of the wheatstone Bridge:
$\dfrac{A}{B}=\dfrac{C}{D}$
Where A, B, C, D are the values of resistances in the bridge circuit.
When this condition is satisfied, no current passes through the galvanometer of the circuit. The condition is satisfied in the problem above, so we neglect the galvanometer from the circuit.

Note:
We must firstly check if the balanced condition of the Wheatstone bridge is satisfied in this case. Only when satisfied, we can proceed to neglect the galvanometer in the circuit and calculate the value of equivalent resistance of the circuit. Conversely, it may be given that the balanced condition is satisfied. In such a case, we can calculate the value of any unknown resistance of any of the given legs of the bridge.