Determination of unknonw resistance in metre bridge is most accurate , when the balancing length is

Options

A. 50 cm
B. 10 cm
C. 20 cm
D. 40 cm

Answer

To determine the unknown resistance in a meter bridge with maximum accuracy, we need to analyze the relationship between the balancing length and the percentage error in the measurement of the unknown resistance. Here’s a step-by-step solution:

### Step 1: Understanding the Meter Bridge
The meter bridge is based on the principle of the Wheatstone bridge. The formula for the unknown resistance \( X \) in terms of the known resistance \( R \) and the balancing length \( L \) is given by:

\[ \frac{X}{R} = \frac{L}{100 - L} \]

### Step 2: Rearranging the Formula
From the above equation, we can express \( X \) as:

\[ X = \frac{L}{100 - L} \times R \]

### Step 3: Differentiating to Find \( dx \)
To find the accuracy of the measurement, we need to determine the change in \( X \) (denoted as \( dx \)) with respect to a change in \( L \) (denoted as \( dL \)). We will differentiate \( X \) with respect to \( L \):

Using the quotient rule for differentiation, we have:

\[ dx = R \left( \frac{(100 - L) \cdot dL - L \cdot (-dL)}{(100 - L)^2} \right) \]

This simplifies to:

\[ dx = R \cdot \frac{(100 - L + L) \cdot dL}{(100 - L)^2} = R \cdot \frac{100 \cdot dL}{(100 - L)^2} \]

### Step 4: Finding the Relative Error
The relative error in the measurement of \( X \) is given by:

\[ \text{Relative Error} = \frac{dx}{X} \]

Substituting for \( dx \) and \( X \):

\[ \text{Relative Error} = \frac{R \cdot \frac{100 \cdot dL}{(100 - L)^2}}{\frac{L}{100 - L} \cdot R} \]

This simplifies to:

\[ \text{Relative Error} = \frac{100 \cdot dL}{L \cdot (100 - L)} \]

### Step 5: Minimizing the Relative Error
To achieve maximum accuracy, we need to minimize the relative error. This can be done by maximizing the denominator \( L \cdot (100 - L) \).

### Step 6: Finding the Maximum
Let \( f(L) = L \cdot (100 - L) = 100L - L^2 \). To find the maximum value, we differentiate \( f(L) \):

\[ f'(L) = 100 - 2L \]

Setting \( f'(L) = 0 \):

\[ 100 - 2L = 0 \implies L = 50 \]

### Step 7: Confirming Maximum
To confirm that this is a maximum, we check the second derivative:

\[ f''(L) = -2 \]

Since \( f''(L) < 0 \), this indicates that \( L = 50 \) cm is indeed a maximum.

### Conclusion
Thus, the determination of unknown resistance in a meter bridge is most accurate when the balancing length is:

\[ \boxed{50 \text{ cm}} \]