A galvanometer of resistance `G` is shunted by a resistance `S ohm`. To keep the main current in the circuit uncharged, the resistnace to be put in series with the galvonmeter

Options

A. `(G)/((S+G))`
B. `(S^(2))/((S+G))`
C. `(SG)/((S+G))`
D. `(G^(2))/((S+G))`

Explanation

To solve the problem, we need to find the resistance \( R \) that should be put in series with the galvanometer to keep the main current in the circuit unchanged when a shunt resistance \( S \) is connected in parallel with the galvanometer of resistance \( G \).

### Step-by-Step Solution:

1. **Understand the Circuit Configuration**:
- We have a galvanometer with resistance \( G \).
- A shunt resistance \( S \) is connected in parallel with the galvanometer.
- A resistance \( R \) is connected in series with the combination of the galvanometer and the shunt.

2. **Calculate the Equivalent Resistance of the Parallel Combination**:
- The equivalent resistance \( R_{eq} \) of the galvanometer and the shunt can be calculated using the formula for resistors in parallel:
\[ R_{eq} = \frac{G \cdot S}{G + S} \]

3. **Total Resistance in the Circuit**:
- The total resistance \( R_{total} \) in the circuit, which includes the series resistance \( R \), is given by:
\[ R_{total} = R + R_{eq} = R + \frac{G \cdot S}{G + S} \]

4. **Setting Up the Equation**:
- To keep the main current in the circuit unchanged, the total resistance after adding \( R \) must equal the resistance \( G \) of the galvanometer alone:
\[ R + \frac{G \cdot S}{G + S} = G \]

5. **Rearranging the Equation**:
- Rearranging gives us:
\[ R = G - \frac{G \cdot S}{G + S} \]

6. **Finding a Common Denominator**:
- To simplify, we can express \( R \) with a common denominator:
\[ R = \frac{G(G + S) - G \cdot S}{G + S} \]
- This simplifies to:
\[ R = \frac{G^2}{G + S} \]

7. **Final Result**:
- Thus, the resistance \( R \) that should be put in series with the galvanometer is:
\[ R = \frac{G^2}{G + S} \]