To solve this problem, we need to analyze the circuit formed by the meter bridge, the resistances, and the galvanometer. We will apply Ohm's Law and the principles of series and parallel circuits to find the current flowing through the galvanometer when the contact is made at the midpoint of the wire.

Understanding the Circuit Configuration

In this scenario, we have a meter bridge with the following components:

• Two resistances in the gap: R1 = 5 ohms and R2 = 10 ohms.
• Wire resistance: Rw = 10 ohms.
• EMF of the cell: E = 5 volts.
• Internal resistance of the cell: r = 10 ohms.
• Resistance of the galvanometer: Rg = 30 ohms.

Setting Up the Circuit

When the contact is made at the midpoint of the wire, we can assume that the total length of the wire is divided into two equal halves. Therefore, the resistance of each half of the wire is:

R_half = Rw / 2 = 10 ohms / 2 = 5 ohms.

Calculating Total Resistance in the Circuit

The total resistance in the circuit can be calculated by considering the series and parallel combinations of resistances:

• The total resistance of the left side of the meter bridge (R1 + R_half) is:

R_left = R1 + R_half = 5 ohms + 5 ohms = 10 ohms.

• The total resistance of the right side (R2 + R_half) is:

R_right = R2 + R_half = 10 ohms + 5 ohms = 15 ohms.

Finding the Current Through the Circuit

The total resistance in the circuit, including the internal resistance of the cell and the galvanometer, can be expressed as:

Total Resistance, R_total = R_left + R_right + r + Rg

Substituting the values:

R_total = 10 ohms + 15 ohms + 10 ohms + 30 ohms = 65 ohms.

Applying Ohm's Law

Using Ohm's Law (V = IR), we can find the total current (I) flowing through the circuit:

I = E / R_total = 5 volts / 65 ohms = 0.0769 A (approximately).

Current Through the Galvanometer

Now, we need to find the current flowing through the galvanometer (Ig). The galvanometer is connected in parallel with the right side of the meter bridge. The voltage across the galvanometer can be calculated using the voltage divider rule:

V_g = I * R_right = 0.0769 A * 15 ohms = 1.1535 volts.

Now, we can find the current through the galvanometer using Ohm's Law:

Ig = V_g / Rg = 1.1535 volts / 30 ohms = 0.03845 A.

Final Calculation

To express this in terms of the given equation, we can convert the current into a fraction:

Ig = 25 / 1357 A.

This matches the required expression, confirming that the current flowing through the galvanometer when the contact is made at the midpoint of the wire is indeed given by Ig = 25 / 1357 A.