To solve the problem step by step, we will follow these calculations:
### Step 1: Understand the Circuit
We have a galvanometer (G) with a resistance of \( R_g = 50 \, \Omega \) connected in series with a resistor \( R_s = 2950 \, \Omega \) and a battery of \( V = 3 \, V \).
### Step 2: Calculate the Initial Current
The total resistance in the circuit is the sum of the galvanometer resistance and the series resistance:
\[
R_{total} = R_g + R_s = 50 \, \Omega + 2950 \, \Omega = 3000 \, \Omega
\]
Using Ohm's law, the initial current \( I_1 \) flowing through the circuit can be calculated as:
\[
I_1 = \frac{V}{R_{total}} = \frac{3 \, V}{3000 \, \Omega} = 0.001 \, A \, (or \, 1 \, mA)
\]
### Step 3: Relate Current to Deflection
The galvanometer shows full-scale deflection (30 divisions) at this current. We can express the deflection \( D \) in terms of the current:
\[
D_1 = k \cdot I_1 \quad \text{(where \( k \) is a constant)}
\]
Thus, we have:
\[
30 = k \cdot I_1 \quad \text{(1)}
\]
### Step 4: Calculate the Current for Reduced Deflection
Now, we want to reduce the deflection to 20 divisions. Let the new resistance in series be \( R \). The new total resistance will be:
\[
R_{new} = R + R_g = R + 50 \, \Omega
\]
The new current \( I_2 \) can be expressed as:
\[
I_2 = \frac{V}{R_{new}} = \frac{3 \, V}{R + 50 \, \Omega}
\]
The new deflection will be:
\[
D_2 = k \cdot I_2 = 20 \quad \text{(2)}
\]
### Step 5: Set Up the Ratio of Deflections
From equations (1) and (2), we can set up the ratio of the currents:
\[
\frac{D_1}{D_2} = \frac{I_1}{I_2}
\]
Substituting the values:
\[
\frac{30}{20} = \frac{I_1}{I_2} \implies \frac{3}{2} = \frac{I_1}{I_2}
\]
Substituting for \( I_1 \) and \( I_2 \):
\[
\frac{3}{2} = \frac{I_1}{\frac{3}{R + 50}} \implies \frac{3}{2} = \frac{3 \cdot (R + 50)}{3} \implies \frac{3}{2} = \frac{R + 50}{3000}
\]
### Step 6: Solve for R
Cross-multiplying gives:
\[
3 \cdot 3000 = 2(R + 50)
\]
\[
9000 = 2R + 100 \implies 2R = 9000 - 100 = 8900 \implies R = \frac{8900}{2} = 4450 \, \Omega
\]
### Final Answer
The resistance in series should be \( R = 4450 \, \Omega \).
