Question

In a meter bridge experiment, a resistance of 9 Ω is connected in the left gap and an unknown resistance greater than 9 Ω is connected in the right gap. If the resistance in the gaps are interchanged, the balancing point shifts by 10 cm. The unknown resistance is:

• A. \( 18 \) Ω
• B. \( 22 \) Ω
• C. \( 11 \) Ω
• D. \( 36 \) Ω

Hint:

In a meter bridge, interchanging the resistances shifts the balancing length. The resistance ratio can be determined using the shift in balance length.

Answer 1

The Correct Option is C

To determine the unknown resistance in the meter bridge experiment, we analyze the given conditions and use the principle of the meter bridge. Given: 
- Resistance in the left gap, \( R_1 = 9 \, \Omega \) 
- Unknown resistance in the right gap, \( R_2>9 \, \Omega \) 
- When the resistances are interchanged, the balancing point shifts by 10 cm. 

Step 1: Initial Balancing Condition In the meter bridge, the balancing condition is: \[ \frac{R_1}{R_2} = \frac{l}{100 - l} \] where \( l \) is the balancing length from the left end. Initially: \[ \frac{9}{R_2} = \frac{l}{100 - l} \]

Step 2: After Interchanging Resistances When the resistances are interchanged, the new balancing condition is: \[ \frac{R_2}{9} = \frac{l + 10}{90 - l} \] Here, the balancing length shifts by 10 cm, so the new balancing length is \( l + 10 \) cm. 

Step 3: Solve the Equations From the initial condition: \[ 9 (100 - l) = R_2 l \] \[ 900 - 9l = R_2 l \] \[ R_2 = \frac{900 - 9l}{l} \] From the interchanged condition: \[ R_2 (90 - l) = 9 (l + 10) \] \[ R_2 = \frac{9 (l + 10)}{90 - l} \] 

Step 4: Equate the Two Expressions for \( R_2 \) \[ \frac{900 - 9l}{l} = \frac{9 (l + 10)}{90 - l} \] 
Simplify: \[ (900 - 9l)(90 - l) = 9l (l + 10) \] \[ 81000 - 900l - 810l + 9l^2 = 9l^2 + 90l \] \[ 81000 - 1710l = 90l \] \[ 81000 = 1800l \] \[ l = \frac{81000}{1800} = 45 \, \text{cm} \] Step 5: Calculate \( R_2 \) Using the initial condition: \[ R_2 = \frac{900 - 9 \times 45}{45} = \frac{900 - 405}{45} = \frac{495}{45} = 11 \, \Omega \] Final Answer: \[ \boxed{11 \, \Omega} \] This corresponds to option (3).