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 current-carrying wires and how the magnetic field changes when one of the currents is turned off.

### Step-by-Step Solution:

1. **Understanding the Setup**:
- We have two parallel wires. Wire 1 carries a current \( I \) and Wire 2 carries a current \( 2I \). Both currents are in the same direction.
- We need to find the magnetic field at the midpoint between these two wires.

2. **Magnetic Field Due to a Long Straight Current-Carrying Wire**:
- The magnetic field \( B \) at a distance \( r \) from a long straight wire carrying current \( I \) is given by:
\[ B = \frac{\mu_0 I}{2\pi r} \]
- For Wire 1 (current \( I \)):
\[ B_1 = \frac{\mu_0 I}{2\pi r} \]
- For Wire 2 (current \( 2I \)):
\[ B_2 = \frac{\mu_0 (2I)}{2\pi r} = \frac{2\mu_0 I}{2\pi r} = \frac{\mu_0 I}{\pi r} \]

3. **Direction of the Magnetic Fields**:
- Using the right-hand rule, the magnetic field due to Wire 1 (\( B_1 \)) at the midpoint will be directed into the plane of the paper.
- The magnetic field due to Wire 2 (\( B_2 \)) will be directed out of the plane of the paper.

4. **Net Magnetic Field at the Midpoint**:
- Since \( B_1 \) is into the plane and \( B_2 \) is out of the plane, the net magnetic field \( B \) at the midpoint is:
\[ B = B_2 - B_1 = \frac{\mu_0 I}{\pi r} - \frac{\mu_0 I}{2\pi r} \]
- Simplifying this:
\[ B = \frac{\mu_0 I}{\pi r} - \frac{\mu_0 I}{2\pi r} = \frac{2\mu_0 I}{2\pi r} - \frac{\mu_0 I}{2\pi r} = \frac{\mu_0 I}{2\pi r} = B \]

5. **Switching Off the Current in Wire 1**:
- When the current \( I \) in Wire 1 is switched off, the magnetic field at the midpoint will only be due to Wire 2:
\[ B' = B_2 = \frac{\mu_0 (2I)}{2\pi r} = \frac{2\mu_0 I}{2\pi r} = 2B \]

6. **Conclusion**:
- Therefore, when the current in Wire 1 is switched off, the magnetic field at the midpoint becomes:
\[ B' = 2B \]

### Final Answer:
The magnetic field will become \( 2B \).

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