In a Wheatstone's brigde all the four arms have equal resistance `R`. If the resistance of the galvanometer arm is also `R`, the equivalent resistance of the combination as seen b the battery is

Options

A. `(R )/(2)`
B. `R`
C. `2 R`
D. `(R )/(4)`

Explanation

To find the equivalent resistance of a Wheatstone bridge where all four arms have equal resistance \( R \) and the galvanometer arm also has resistance \( R \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Configuration**:
In a Wheatstone bridge, we have four resistances arranged in a diamond shape. Let’s label them as follows:
- \( R_1 = R \) (top arm)
- \( R_2 = R \) (bottom arm)
- \( R_3 = R \) (left arm)
- \( R_4 = R \) (right arm)
- \( R_g = R \) (galvanometer arm)

2. **Check for Balance**:
The Wheatstone bridge is balanced when the ratio of the resistances in one branch is equal to the ratio in the other branch. For our case:
\[ \frac{R_1}{R_2} = \frac{R_3}{R_4} \]
Since all resistances are equal (\( R \)), this condition simplifies to:
\[ \frac{R}{R} = \frac{R}{R} \implies 1 = 1 \]
Thus, the bridge is balanced.

3. **Effect of Balance**:
When the bridge is balanced, no current flows through the galvanometer. Therefore, we can ignore the galvanometer in our calculations, effectively removing it from the circuit.

4. **Redraw the Circuit**:
After removing the galvanometer, we are left with two pairs of resistors in parallel:
- The top and bottom resistors (\( R_1 \) and \( R_2 \)) are in series, and their total resistance is:
\[ R_{top} = R + R = 2R \]
- The left and right resistors (\( R_3 \) and \( R_4 \)) are also in series, giving:
\[ R_{bottom} = R + R = 2R \]

5. **Calculate the Equivalent Resistance**:
Now, we have two resistances \( R_{top} \) and \( R_{bottom} \) in parallel:
\[ R_{eq} = \frac{R_{top} \times R_{bottom}}{R_{top} + R_{bottom}} = \frac{2R \times 2R}{2R + 2R} \]
Simplifying this:
\[ R_{eq} = \frac{4R^2}{4R} = R \]

### Final Answer:
The equivalent resistance of the combination as seen by the battery is \( R \).

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