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

Explanation

To solve the problem of finding the resistance to be put in series with a galvanometer of resistance \( G \) that is shunted by a resistance \( S \) ohm, we can follow these steps:

### Step 1: Understand the Circuit Configuration
We have a galvanometer with a resistance \( G \) and a shunt resistance \( S \) connected in parallel to it. The total current flowing through the circuit should remain unchanged when the shunt is added.

### Step 2: Apply the Concept of Equivalent Resistance
When the galvanometer and the shunt are connected in parallel, the equivalent resistance \( R_{eq} \) of the parallel combination can be calculated using the formula:
\[ \frac{1}{R_{eq}} = \frac{1}{G} + \frac{1}{S} \]
This formula allows us to find the total resistance of the parallel circuit.

### Step 3: Calculate the Equivalent Resistance
Rearranging the formula gives us:
\[ R_{eq} = \frac{GS}{G + S} \]

### Step 4: Determine the Total Resistance in the Circuit
Let \( R \) be the resistance that we need to add in series with the galvanometer. The total resistance \( R_{total} \) in the circuit will then be:
\[ R_{total} = R + R_{eq} \]

### Step 5: Set Up the Condition for Unchanged Current
To keep the main current in the circuit unchanged, the total resistance must remain equal to the original resistance \( R_{original} \) of the circuit without the shunt. Thus, we can write:
\[ R + R_{eq} = R_{original} \]

### Step 6: Solve for the Series Resistance \( R \)
Substituting \( R_{eq} \) into the equation gives:
\[ R + \frac{GS}{G + S} = R_{original} \]
From this, we can solve for \( R \):
\[ R = R_{original} - \frac{GS}{G + S} \]

### Final Expression
Thus, the resistance to be put in series with the galvanometer is:
\[ R = R_{original} - \frac{GS}{G + S} \]