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: Set up the initial conditions
We have a meter bridge with a total length of 100 cm. A resistance of \( R_1 = 2 \Omega \) is connected in one gap, and an unknown resistance \( R \) (greater than \( 2 \Omega \)) is connected in the other gap. Let the balance point be at a distance \( x \) from point A.

### Step 2: Write the balance condition for the first configuration
Using the principle of the meter bridge, we can write the balance condition as:
\[ \frac{R_1}{x} = \frac{R}{100 - x} \]
Substituting \( R_1 = 2 \Omega \):
\[ \frac{2}{x} = \frac{R}{100 - x} \]
Cross-multiplying gives:
\[ 2(100 - x) = Rx \]
This simplifies to:
\[ 200 - 2x = Rx \quad \text{(Equation 1)} \]

### Step 3: Set up the conditions after interchanging the resistances
When the resistances are interchanged, \( R \) is now in the first gap and \( 2 \Omega \) is in the second gap. The balance point shifts by 20 cm, so the new balance point is at \( x + 20 \) cm from A. The new balance condition is:
\[ \frac{R}{x + 20} = \frac{2}{80 - x} \]
Cross-multiplying gives:
\[ R(80 - x) = 2(x + 20) \]
This simplifies to:
\[ R(80 - x) = 2x + 40 \quad \text{(Equation 2)} \]

### Step 4: Solve the two equations
From Equation 1, we can express \( R \):
\[ R = \frac{200 - 2x}{x} \]

Substituting this expression for \( R \) into Equation 2:
\[ \frac{200 - 2x}{x}(80 - x) = 2x + 40 \]
Cross-multiplying gives:
\[ (200 - 2x)(80 - x) = x(2x + 40) \]
Expanding both sides:
\[ 16000 - 200x - 160x + 2x^2 = 2x^2 + 40x \]
This simplifies to:
\[ 16000 - 360x = 40x \]
Combining like terms:
\[ 16000 = 400x \]
Thus,
\[ x = \frac{16000}{400} = 40 \text{ cm} \]

### Step 5: Substitute back to find \( R \)
Now substituting \( x = 40 \) cm back into the expression for \( R \):
\[ R = \frac{200 - 2(40)}{40} = \frac{200 - 80}{40} = \frac{120}{40} = 3 \Omega \]

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

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