The magnetic field midway between two parallel current carrying wires, carrying currents I and 2i in same direction is B. If the current in the wire with current I is switched off, the magnetic field will become

Options

A. `B//3`
B. `2B`
C. `B//2`
D. `B//4`

Answer

To solve the problem, we need to analyze the magnetic fields produced by two parallel wires carrying currents \( I \) and \( 2I \) in the same direction.

### Step-by-Step Solution:

1. **Identify the Magnetic Fields**:
- For a long straight wire, the magnetic field at a distance \( r \) from the wire carrying current \( I \) is given by the formula:
\[ B = \frac{\mu_0 I}{2\pi r} \]
- For the first wire carrying current \( I \):
\[ B_1 = \frac{\mu_0 I}{2\pi r} \]
- For the second wire carrying current \( 2I \):
\[ B_2 = \frac{\mu_0 (2I)}{2\pi r} = \frac{\mu_0 2I}{2\pi r} \]

2. **Determine the Direction of the Magnetic Fields**:
- Using the right-hand rule, the magnetic field \( B_1 \) due to the first wire (current \( I \)) will be directed downwards at the midpoint between the two wires.
- The magnetic field \( B_2 \) due to the second wire (current \( 2I \)) will be directed upwards at the same point.

3. **Calculate the Net Magnetic Field**:
- Since \( B_2 \) is greater than \( B_1 \), the net magnetic field \( B \) at the midpoint is:
\[ B = B_2 - B_1 = \frac{\mu_0 2I}{2\pi r} - \frac{\mu_0 I}{2\pi r} = \frac{\mu_0 I}{2\pi r} \]
- Thus, the net magnetic field \( B \) is:
\[ B = \frac{\mu_0 I}{2\pi r} \]

4. **Switch Off the Current in the First Wire**:
- When the current \( I \) in the first wire is switched off, the magnetic field due to the first wire becomes zero.
- The magnetic field at the midpoint is now only due to the second wire with current \( 2I \):
\[ B' = B_2 = \frac{\mu_0 (2I)}{2\pi r} \]

5. **Express the New Magnetic Field in Terms of B**:
- We previously found that:
\[ B = \frac{\mu_0 I}{2\pi r} \]
- Therefore, the new magnetic field \( B' \) can be expressed as:
\[ B' = 2 \times \frac{\mu_0 I}{2\pi r} = 2B \]

### Conclusion:
When the current in the wire carrying current \( I \) is switched off, the magnetic field at the midpoint becomes \( 2B \).