Resistance in the two gaps of a meter bridge are `10 ohm` and `30 ohm` respectively. If the resistances are interchanged the balance point shifts by

Options

A. `33.3 cm`
B. `66.67 cm`
C. `25 cm`
D. `50 cm`

Explanation

To solve the problem step by step, we will analyze the situation of a meter bridge with two resistances and determine how the balance point shifts when the resistances are interchanged.

### Step-by-Step Solution:

1. **Understanding the Setup**:
We have a meter bridge with two resistances, \( R_1 = 10 \, \Omega \) and \( R_2 = 30 \, \Omega \). The bridge is balanced when the ratio of the resistances is equal to the ratio of the lengths on the bridge.

2. **Setting Up the Balance Condition**:
The balance condition for the meter bridge can be expressed as:
\[ \frac{R_1}{R_2} = \frac{L_1}{100 - L_1} \]
where \( L_1 \) is the length from one end of the bridge to the balance point.

3. **Substituting the Values**:
Substituting \( R_1 = 10 \, \Omega \) and \( R_2 = 30 \, \Omega \) into the balance condition:
\[ \frac{10}{30} = \frac{L_1}{100 - L_1} \]
Simplifying the left side:
\[ \frac{1}{3} = \frac{L_1}{100 - L_1} \]

4. **Cross-Multiplying**:
Cross-multiplying gives:
\[ 100 - L_1 = 3L_1 \]

5. **Solving for \( L_1 \)**:
Rearranging the equation:
\[ 100 = 4L_1 \implies L_1 = \frac{100}{4} = 25 \, \text{cm} \]

6. **Interchanging the Resistances**:
Now, we interchange the resistances, so \( R_1 = 30 \, \Omega \) and \( R_2 = 10 \, \Omega \). We set up the new balance condition:
\[ \frac{R_1}{R_2} = \frac{L_2}{100 - L_2} \]
Substituting the new values:
\[ \frac{30}{10} = \frac{L_2}{100 - L_2} \]
Simplifying:
\[ 3 = \frac{L_2}{100 - L_2} \]

7. **Cross-Multiplying Again**:
Cross-multiplying gives:
\[ 100 - L_2 = \frac{L_2}{3} \]

8. **Solving for \( L_2 \)**:
Rearranging the equation:
\[ 300 - 3L_2 = L_2 \implies 300 = 4L_2 \implies L_2 = \frac{300}{4} = 75 \, \text{cm} \]

9. **Calculating the Shift in Balance Point**:
The shift in the balance point is calculated by finding the difference between the new and old balance points:
\[ \text{Shift} = L_2 - L_1 = 75 \, \text{cm} - 25 \, \text{cm} = 50 \, \text{cm} \]

### Final Answer:
The balance point shifts by **50 cm**.