**Introduction**

The half deflection method is a technique used to determine the resistance of a galvanometer, which measures small electric currents. This method involves connecting the galvanometer in series with a known resistance and a variable resistance. By adjusting the variable resistance until the galvanometer deflects by half of its maximum value, the resistance of the galvanometer can be calculated.

**Given Data**

- R1 = 9970 Ω (known resistance)
- R2 = 30 Ω (variable resistance)
- R3 = 0 Ω (initially)
- R3 = 107 Ω (changed resistance)
- d = initial deflection
- d2 = deflection after changing R3

**Procedure**

1. Connect the galvanometer in series with R1, R2, and R3.
2. Initially, set R3 to 0 Ω.
3. Adjust R2 until the galvanometer deflects by half of its maximum value (d).
4. Note down the value of R2 (let's call it R2_initial).
5. Change R3 to 107 Ω.
6. Adjust R2 again until the galvanometer deflects by half of its maximum value (d2).
7. Note down the value of R2 (let's call it R2_changed).

**Calculations**

1. The resistance of the galvanometer can be calculated using the formula:

Rg = (R2_initial - R2_changed) / (R2_changed - R1) * R1

2. Substitute the given values:

Rg = (R2_initial - R2_changed) / (R2_changed - 9970) * 9970

**Explanation**

The resistance of the galvanometer is determined by comparing the changes in R2 required to achieve half deflection in the galvanometer at two different values of R3. When R3 is 0 Ω, the deflection is d, and when R3 is 107 Ω, the deflection changes to d2.

By adjusting R2 until the galvanometer deflects by half of its maximum value, we can find the balance point where the current through the galvanometer is equal to half of the maximum current. This balance point is achieved when the resistance of the galvanometer is equal to the total resistance in the circuit.

By comparing the changes in R2 required to achieve half deflection at two different values of R3, we can calculate the resistance of the galvanometer using the formula mentioned above. This formula utilizes the relationship between the resistance and the deflection of the galvanometer.

**Conclusion**

Using the half deflection method, we can determine the resistance of a galvanometer by comparing the changes in the variable resistance required to achieve half deflection at different values of a known resistance. The resistance of the galvanometer can be calculated using the formula mentioned above.

It's 77 ohm