Consider the circuit shown. Galvanometer resistance is 10 Ω and current through galvanometer is 3 mA. Find the resistance of shunt.

The correct option is 3.75 × 10^–3 Ω.

1. Understanding the Circuit from the Diagram:
From the given circuit diagram, we can observe that a total current labeled as $i=8\text{ A}$ enters the junction. The circuit then splits into two parallel branches:
- The upper branch contains a shunt resistor with resistance $S$.
- The lower branch contains a galvanometer with resistance $G$.

2. Identifying the Given Parameters:
- Galvanometer resistance, $G=10\text{ Ω}$
- Current through the galvanometer, $i_g=3\text{ mA}=3\times {10}^{-3}\text{ A}$
- Total current, $i=8\text{ A}$

3. Formulation:
In a parallel combination, the potential difference (voltage) across the shunt resistor is equal to the potential difference across the galvanometer:

$V_s=V_g$

$i_s\times S=i_g\times G$

where $i_s$ is the current flowing through the shunt resistor. By Kirchhoff's Current Law, the remaining current flows through the shunt:

$i_s=i-i_g$

Substituting this expression into the voltage equation gives:

$(i-i_g)\times S=i_g\times G$

Rearranging the equation to solve for the shunt resistance $S$:

$S=\frac{i_g\times G}{i-i_g}$

4. Step-by-Step Calculation:
Substitute the values into the formula:

$S=\frac{(3\times {10}^{-3}\text{ A})\times 10\text{ Ω}}{8\text{ A}-3\times {10}^{-3}\text{ A}}$

Simplify the numerator and the denominator:

$S=\frac{3\times {10}^{-2}}{8-0.003}$

$S=\frac{0.03}{7.997}$

$S\approx 0.0037514\text{ Ω}=3.75\times {10}^{-3}\text{ Ω}$