Show that the percentage error in the measurement of resistance by a metre bridge is minimum when the null point is near about the centre of the wire.

Options

A. 20 cm
B. 50 cm
C. 80 cm
D. 100 cm

Answer

To show that the percentage error in the measurement of resistance by a metre bridge is minimum when the null point is near about the center of the wire, we can follow these steps:

### Step 1: Define the variables
Let:
- \( R \) = the resistance to be measured
- \( L \) = the length of the wire from one end to the null point
- The total length of the wire = 100 cm

### Step 2: Establish the relationship using the metre bridge principle
According to the principle of the metre bridge, we have:
\[ \frac{x}{R} = \frac{L}{100 - L} \]
Where \( x \) is the resistance in the segment of the wire corresponding to the length \( L \).

### Step 3: Rearranging the equation
From the above equation, we can express \( x \) as:
\[ x = R \cdot \frac{L}{100 - L} \]

### Step 4: Calculate the differential error in \( x \)
To find the error in the measurement of resistance \( x \), we differentiate \( x \):
\[ \Delta x = R \cdot \left( \frac{\Delta L (100 - L) - L \Delta L}{(100 - L)^2} \right) \]
This simplifies to:
\[ \Delta x = R \cdot \frac{\Delta L \cdot (100 - 2L)}{(100 - L)^2} \]

### Step 5: Calculate the relative error in \( x \)
The relative error in \( x \) is given by:
\[ \frac{\Delta x}{x} = \frac{\Delta L \cdot (100 - 2L)}{L(100 - L)} \]

### Step 6: Express the percentage error
To express this as a percentage error, we multiply by 100:
\[ \text{Percentage Error} = \frac{\Delta L \cdot (100 - 2L)}{L(100 - L)} \cdot 100 \]

### Step 7: Minimize the percentage error
To minimize the percentage error, we need to maximize the term \( L(100 - L) \). This is a quadratic function that achieves its maximum value when:
\[ L = 100 - L \implies 2L = 100 \implies L = 50 \]

### Conclusion
Thus, the percentage error in the measurement of resistance by a metre bridge is minimum when the null point is at the center of the wire, i.e., \( L = 50 \) cm.