A resistance box, a battery and a galvanometer of resistance `G` ohm are connected in series. the galvanometer is shunted by resistance of `S` ohm, find the change in resistance in the be required to maintain the current from the battery unchanged.

Explanation

To solve the problem step-by-step, we will analyze the circuit and apply the principles of series and parallel resistances.

### Step 1: Understand the Circuit Configuration
We have a battery, a resistance box (which we will denote as R), and a galvanometer with resistance G ohms connected in series. The galvanometer is shunted by a resistance S ohms.

### Step 2: Identify the Equivalent Resistance
Since the galvanometer and the shunt resistance S are in parallel, we need to find the equivalent resistance (R_eq) of the galvanometer and the shunt resistance. The formula for the equivalent resistance of two resistors in parallel is given by:

\[ \frac{1}{R_{eq}} = \frac{1}{G} + \frac{1}{S} \]

### Step 3: Calculate the Equivalent Resistance
Rearranging the formula for R_eq, we have:

\[ R_{eq} = \frac{GS}{G + S} \]

### Step 4: Total Resistance in the Circuit
The total resistance (R_total) in the circuit will be the sum of the resistance of the resistance box (R) and the equivalent resistance of the galvanometer and shunt:

\[ R_{total} = R + R_{eq} = R + \frac{GS}{G + S} \]

### Step 5: Condition for Unchanged Current
For the current from the battery to remain unchanged, the total resistance must remain constant. Therefore, if we denote the original total resistance as R_total_initial, we can set up the equation:

\[ R + \frac{GS}{G + S} = R_{total\_initial} \]

### Step 6: Solve for Change in Resistance
To find the change in resistance (ΔR), we need to isolate R:

\[ R = R_{total\_initial} - \frac{GS}{G + S} \]

### Step 7: Express the Change in Resistance
If we denote the original resistance (before shunting) as R_initial, we can express the change in resistance required to maintain the current unchanged as:

\[ \Delta R = R_{initial} - \left(R_{total\_initial} - \frac{GS}{G + S}\right) \]

### Final Result
After simplifying, we find that:

\[ \Delta R = \frac{G^2}{G + S} \]

This is the change in resistance required to maintain the current from the battery unchanged.

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