In a meter bridge when the resistance in the left gap is `4 Omega` and an unknown resistance in the right gap the balance point is obtained at `40 cm` from the zero end. On shunting the unknown resistance with `4 Omega`, find the shift of the balance point on the bridge wire

Explanation

To solve the problem step by step, we will follow the principles of a meter bridge and the concept of parallel resistances.

### Step 1: Understand the Initial Setup
In the meter bridge, we have:
- Resistance in the left gap (R1) = 4 Ω
- Unknown resistance in the right gap (R2) = y (unknown)
- Balance point (L) = 40 cm from the zero end

### Step 2: Apply the Balance Condition
The balance condition in a meter bridge can be expressed as:
\[ \frac{R_1}{L} = \frac{R_2}{100 - L} \]
Substituting the known values:
\[ \frac{4}{40} = \frac{y}{60} \]

### Step 3: Solve for the Unknown Resistance (y)
Cross-multiplying gives:
\[ 4 \times 60 = 40 \times y \]
\[ 240 = 40y \]
\[ y = \frac{240}{40} = 6 \, \Omega \]

### Step 4: Shunt the Unknown Resistance
Now, we shunt the unknown resistance (6 Ω) with an additional 4 Ω resistance. The equivalent resistance (R_eq) of two resistances in parallel is given by:
\[ \frac{1}{R_{eq}} = \frac{1}{y} + \frac{1}{4} \]
Substituting the value of y:
\[ \frac{1}{R_{eq}} = \frac{1}{6} + \frac{1}{4} \]

### Step 5: Calculate the Equivalent Resistance
Finding a common denominator (12):
\[ \frac{1}{R_{eq}} = \frac{2}{12} + \frac{3}{12} = \frac{5}{12} \]
Thus,
\[ R_{eq} = \frac{12}{5} = 2.4 \, \Omega \]

### Step 6: Find the New Balance Point
Now we need to find the new balance point (L') using the same balance condition:
\[ \frac{R_1}{L'} = \frac{R_{eq}}{100 - L'} \]
Substituting the known values:
\[ \frac{4}{L'} = \frac{2.4}{100 - L'} \]

### Step 7: Solve for the New Balance Point (L')
Cross-multiplying gives:
\[ 4(100 - L') = 2.4L' \]
Expanding this:
\[ 400 - 4L' = 2.4L' \]
Combining like terms:
\[ 400 = 4L' + 2.4L' \]
\[ 400 = 6.4L' \]
Thus,
\[ L' = \frac{400}{6.4} = 62.5 \, \text{cm} \]

### Step 8: Calculate the Shift in the Balance Point
The shift in the balance point is given by:
\[ \text{Shift} = L' - L = 62.5 \, \text{cm} - 40 \, \text{cm} = 22.5 \, \text{cm} \]

### Final Answer
The shift of the balance point on the bridge wire is **22.5 cm**.

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