A galvanometer coil of resistance `50 Omega`, show full deflection of `100 muA` . The shunt resistance to be added to the galvanometer, to work as an ammeter of range 10 mA is

Options

A. `5 Omega` in parallel
B. `0.5 Omega` in series
C. `5 Omega` in series
D. `0.5 Omega` in parallel

Answer

To find the shunt resistance \( R \) that needs to be added to the galvanometer in order to convert it into an ammeter with a range of 10 mA, we can follow these steps:

### Step 1: Identify Given Values
- Resistance of the galvanometer, \( R_g = 50 \, \Omega \)
- Full-scale deflection current of the galvanometer, \( I_g = 100 \, \mu A = 100 \times 10^{-6} \, A \)
- Desired maximum current for the ammeter, \( I = 10 \, mA = 10 \times 10^{-3} \, A \)

### Step 2: Determine the Current through the Shunt
The current through the shunt resistance \( R \) can be calculated as:
\[ I_R = I - I_g \]
Substituting the values:
\[ I_R = 10 \times 10^{-3} \, A - 100 \times 10^{-6} \, A = 10 \times 10^{-3} \, A - 0.1 \times 10^{-3} \, A = 9.9 \times 10^{-3} \, A \]

### Step 3: Apply the Voltage Equality Condition
Since the galvanometer and the shunt are in parallel, the voltage across both must be equal:
\[ V_g = V_R \]
The voltage across the galvanometer is given by:
\[ V_g = I_g \cdot R_g = (100 \times 10^{-6} \, A) \cdot (50 \, \Omega) = 5 \times 10^{-3} \, V \]
The voltage across the shunt resistance is:
\[ V_R = I_R \cdot R \]
Setting these equal gives:
\[ I_g \cdot R_g = I_R \cdot R \]

### Step 4: Substitute Known Values
Substituting the known values into the equation:
\[ (100 \times 10^{-6} \, A) \cdot (50 \, \Omega) = (9.9 \times 10^{-3} \, A) \cdot R \]
This simplifies to:
\[ 5 \times 10^{-3} \, V = (9.9 \times 10^{-3} \, A) \cdot R \]

### Step 5: Solve for Shunt Resistance \( R \)
Rearranging the equation to solve for \( R \):
\[ R = \frac{5 \times 10^{-3} \, V}{9.9 \times 10^{-3} \, A} \]
Calculating \( R \):
\[ R \approx \frac{5}{9.9} \, \Omega \approx 0.505 \, \Omega \]

### Final Answer
The shunt resistance \( R \) that needs to be added to the galvanometer is approximately:
\[ R \approx 0.51 \, \Omega \]