Resistance of `12 Omega and X Omega` are connected in parallel in the left gap and resistances of `9 Omega and 7 Omega` are connected in series in the right gap of the meter bridge. If the balancing length is 36 cm, then the value of resistance X is

Options

A. `72 Omega`
B. `54 Omega`
C. `36 Omega`
D. `64 Omega`

Explanation

To solve the problem step by step, we will follow the principles of the meter bridge and the rules for resistances in series and parallel.

### Step 1: Understand the Configuration
We have two resistances (12Ω and XΩ) connected in parallel on the left side of the meter bridge, and two resistances (9Ω and 7Ω) connected in series on the right side. The balancing length is given as 36 cm.

### Step 2: Calculate the Total Resistance on the Right Side
The total resistance on the right side (R_right) is the sum of the two resistances in series:
\[ R_{\text{right}} = 9Ω + 7Ω = 16Ω \]

### Step 3: Determine the Lengths on the Meter Bridge
The total length of the meter bridge wire is 100 cm. Since the balancing length is 36 cm, the length on the right side (L_right) can be calculated as:
\[ L_{\text{right}} = 100 cm - 36 cm = 64 cm \]

### Step 4: Use the Balancing Condition of the Meter Bridge
According to the principle of the meter bridge, the ratio of the resistances is equal to the ratio of the lengths:
\[ \frac{R_{\text{left}}}{R_{\text{right}}} = \frac{L_{\text{left}}}{L_{\text{right}}} \]
Substituting the known values:
\[ \frac{R_{\text{left}}}{16Ω} = \frac{36 cm}{64 cm} \]

### Step 5: Calculate R_left
Now, we can express R_left in terms of the known quantities:
\[ R_{\text{left}} = 16Ω \times \frac{36 cm}{64 cm} \]
Calculating this gives:
\[ R_{\text{left}} = 16Ω \times \frac{36}{64} = 9Ω \]

### Step 6: Calculate the Equivalent Resistance of the Left Side
The equivalent resistance of the left side (R_left) with resistances 12Ω and XΩ in parallel is given by:
\[ \frac{1}{R_{\text{left}}} = \frac{1}{12Ω} + \frac{1}{XΩ} \]
Substituting R_left = 9Ω:
\[ \frac{1}{9} = \frac{1}{12} + \frac{1}{X} \]

### Step 7: Solve for X
Rearranging the equation:
\[ \frac{1}{X} = \frac{1}{9} - \frac{1}{12} \]
Finding a common denominator (36):
\[ \frac{1}{X} = \frac{4 - 3}{36} = \frac{1}{36} \]
Thus, we find:
\[ X = 36Ω \]

### Conclusion
The value of the unknown resistance \( X \) is \( 36Ω \).

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