The resistance of a galvanometer coil is R. What is the shunt resistance required to convert it into an ammeter of range 4 times

Options

A. R/5
B. R/4
C. R/3
D. 4R

Explanation

To solve the problem of finding the shunt resistance required to convert a galvanometer into an ammeter with a range that is 4 times the original, we can follow these steps:

### Step 1: Understand the Concept
A galvanometer can be converted into an ammeter by connecting a shunt resistance (R') in parallel with it. The purpose of the shunt is to allow most of the current to bypass the galvanometer, thus preventing it from being damaged by excessive current.

### Step 2: Define the Variables
Let:
- \( R \) = resistance of the galvanometer
- \( I_0 \) = full-scale deflection current of the galvanometer
- \( I' \) = current through the ammeter after conversion
- \( I' = 4 I_0 \) (since the range is 4 times)

### Step 3: Apply Ohm's Law
The voltage across the galvanometer and the shunt resistance must be the same when they are connected in parallel. Thus, we can write:
\[ V = I_0 \times R = I' \times R_{eq} \]
where \( R_{eq} \) is the equivalent resistance of the galvanometer and the shunt.

### Step 4: Calculate the Equivalent Resistance
The equivalent resistance \( R_{eq} \) for resistors in parallel is given by:
\[ \frac{1}{R_{eq}} = \frac{1}{R} + \frac{1}{R'} \]
Rearranging gives us:
\[ R_{eq} = \frac{R \cdot R'}{R + R'} \]

### Step 5: Set Up the Equation
Substituting \( I' = 4 I_0 \) into the voltage equation gives:
\[ I_0 \times R = 4 I_0 \times R_{eq} \]
Cancelling \( I_0 \) (assuming \( I_0 \neq 0 \)):
\[ R = 4 R_{eq} \]

### Step 6: Substitute for \( R_{eq} \)
Substituting the expression for \( R_{eq} \):
\[ R = 4 \left( \frac{R \cdot R'}{R + R'} \right) \]

### Step 7: Solve for \( R' \)
Cross-multiplying gives:
\[ R(R + R') = 4R \cdot R' \]
Expanding and rearranging:
\[ R^2 + R R' = 4 R R' \]
\[ R^2 = 4 R R' - R R' \]
\[ R^2 = 3 R R' \]
Dividing both sides by \( R \) (assuming \( R \neq 0 \)):
\[ R = 3 R' \]
Thus, we find:
\[ R' = \frac{R}{3} \]

### Final Answer
The shunt resistance required to convert the galvanometer into an ammeter of range 4 times is:
\[ R' = \frac{R}{3} \]

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