A teacher in his physics laboratory allotted an experiment to determine the resistance $$G$$ a galvanometer. Students took the observations for $$\frac{1}{3}$$ deflection in the galvanometer. Which of the below is true for measuring value of $$G$$?

We need to determine the relation between galvanometer resistance $$G$$ and shunt resistance $$S$$ when using the $$\frac{1}{3}$$ deflection method.

In this method, the full deflection $$\theta$$ is first noted when the galvanometer is connected to a battery through a high resistance. Then a shunt $$S$$ is connected across the galvanometer and adjusted until the deflection reduces to $$\theta/3$$. Since deflection is proportional to the galvanometer current, reducing deflection to $$\theta/3$$ means the current becomes $$I_g/3$$, where $$I_g$$ is the original current. The total current $$I$$ from the source remains approximately $$I_g$$, assuming the series resistance is much larger than both $$G$$ and $$S$$.

Under these conditions, the current through the shunt is $$I_g - I_g/3 = 2I_g/3$$, and the voltage across both branches must be equal, so

$$G \times \frac{I_g}{3} = S \times \frac{2I_g}{3}$$

Solving for $$G$$ gives $$G = \frac{S \times \frac{2I_g}{3}}{\frac{I_g}{3}} = 2S$$, showing that the galvanometer resistance is twice the shunt resistance.

The correct answer is Option B: $$\frac{1}{3}$$ deflection method can be used and in this case $$G$$ equals twice the value of the shunt resistance.