A resistance of `2Omega` is connected across one gap of a meter bridge (the length of the wire is `100 cm`) and an unknown resistance, greater than `2Omega`is conneted across the other gap. When these resistances are interchanged, the balance point shifts by `20 cm`. Neglecting any corrections,the unknown resistance is

Options

A. `3Omega`
B. `4Omega`
C. `5Omega`
D. `6Omega`

Explanation

To solve the problem, we will follow these steps:

### Step 1: Understand the Meter Bridge Setup
A meter bridge consists of a wire of length 100 cm, and we have a known resistance (2Ω) connected across one gap and an unknown resistance (R) connected across the other gap. The balance point is where the galvanometer shows zero current.

### Step 2: Set Up the Balance Condition
When the resistances are in their initial positions, the balance condition can be expressed as:
\[ \frac{2}{X} = \frac{R}{100 - X} \]
where \(X\) is the initial balance length.

### Step 3: Rearranging the Equation
From the balance condition, we can rearrange it to find R:
\[ 2(100 - X) = R \cdot X \]
\[ R = \frac{2(100 - X)}{X} \]

### Step 4: Analyze the Shift in Balance Point
When the resistances are interchanged, the balance point shifts by 20 cm. The new balance point is \(X + 20\). The new balance condition is:
\[ \frac{R}{X + 20} = \frac{2}{80 - X} \]
where \(80 - X\) is the new length on the opposite side.

### Step 5: Rearranging the New Balance Condition
From the new balance condition, we can rearrange it:
\[ R(80 - X) = 2(X + 20) \]

### Step 6: Substitute R from Step 3 into Step 5
Substituting the expression for R from Step 3 into the new balance condition:
\[ \frac{2(100 - X)}{X}(80 - X) = 2(X + 20) \]

### Step 7: Simplifying the Equation
Cancelling 2 from both sides:
\[ \frac{(100 - X)(80 - X)}{X} = X + 20 \]
Cross-multiplying gives:
\[ (100 - X)(80 - X) = X(X + 20) \]

### Step 8: Expanding Both Sides
Expanding both sides:
\[ 8000 - 100X - 80X + X^2 = X^2 + 20X \]
This simplifies to:
\[ 8000 - 180X = 20X \]

### Step 9: Solving for X
Combining like terms:
\[ 8000 = 200X \]
\[ X = \frac{8000}{200} = 40 \text{ cm} \]

### Step 10: Finding the Unknown Resistance R
Now substituting \(X = 40\) cm back into the equation for R:
\[ R = \frac{2(100 - 40)}{40} = \frac{2 \times 60}{40} = \frac{120}{40} = 3 \Omega \]

### Final Answer
The unknown resistance \(R\) is \(3 \Omega\).

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