Explanation:

Wheatstone Bridge:

- A Wheatstone bridge is a circuit used to measure an unknown electrical resistance by balancing two legs of a bridge circuit, one leg of which includes the unknown component.
- The Wheatstone bridge consists of four resistors connected in the form of a diamond as shown in the below figure.


- The bridge is said to be balanced when the potential difference between points C and D is zero.
- At balance, the product of the resistances in the two legs of the bridge across the diagonal are equal. That is,

R1R3 = R2R4

- Here, R1, R2, R3, and R4 are the resistances of the four arms of the Wheatstone bridge.

Given:

- In the given Wheatstone bridge, all four arms have equal resistance R.
- The resistance of the galvanometer arm is also R.

To find:

- The equivalent resistance of the combination as seen by the battery.

Solution:

- Let's assume that the equivalent resistance of the combination is Req.
- At balance, the potential difference between points C and D is zero.
- This implies that the current passing through the galvanometer arm is zero.
- Therefore, the current passing through the resistors R and R is the same.
- Using Ohm's law, we can write the current passing through each resistor as:

I = V/R

- Here, V is the potential difference across the Wheatstone bridge.
- The potential difference across the resistors R and R is V/2.
- Therefore, the current passing through each resistor is given by:

I1 = I2 = (V/2)/R = V/2R

- The potential difference across the galvanometer arm is also V/2.
- Since the resistance of the galvanometer arm is R, the current passing through it is given by:

Ig = (V/2)/R = V/2R

- At balance, the current passing through the galvanometer arm is zero.
- Therefore, the potential difference across the galvanometer arm is also zero.
- This implies that the potential difference across the resistors R and R is also zero.
- Therefore, the potential difference across the two parallel resistors is zero.
- This implies that the equivalent resistance of the two parallel resistors is zero.
- Therefore, the equivalent resistance of the combination is given by:

Req = R + 0 + R = 2R

- Therefore, the correct option is (c) R.