Two parallel wires carrying current I and 2I in same direction have magnetic field B at the midpoint between them. If the current 2I is switched off, the magnetic field at that point will be

Options

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

Answer

To solve the problem, we will analyze the magnetic field produced by two parallel wires carrying currents \( I \) and \( 2I \) in the same direction, and then determine the effect of switching off the current in the second wire.

### Step-by-Step Solution:

1. **Understanding the Setup**:
- We have two parallel wires separated by a distance \( 2a \).
- Wire 1 carries a current \( I \) and Wire 2 carries a current \( 2I \).
- We are interested in the magnetic field at the midpoint between the two wires.

2. **Magnetic Field Due to a Current-Carrying Wire**:
- The magnetic field \( B \) at a distance \( r \) from a long straight wire carrying current \( I \) is given by the formula:
\[ B = \frac{\mu_0 I}{2\pi r} \]
- Here, \( \mu_0 \) is the permeability of free space.

3. **Calculating the Magnetic Field at the Midpoint**:
- The distance from each wire to the midpoint is \( a \) (since the total distance is \( 2a \)).
- The magnetic field due to Wire 1 (current \( I \)) at the midpoint:
\[ B_1 = \frac{\mu_0 I}{2\pi a} \]
- The magnetic field due to Wire 2 (current \( 2I \)) at the midpoint:
\[ B_2 = \frac{\mu_0 (2I)}{2\pi a} = \frac{\mu_0 I}{\pi a} \]

4. **Direction of the Magnetic Fields**:
- For Wire 1 (current \( I \)), the magnetic field at the midpoint will be directed into the page (using the right-hand rule).
- For Wire 2 (current \( 2I \)), the magnetic field at the midpoint will also be directed into the page.

5. **Net Magnetic Field at the Midpoint**:
- Since both magnetic fields are in the same direction (into the page), the net magnetic field \( B \) at the midpoint is:
\[ B = B_1 + B_2 = \frac{\mu_0 I}{2\pi a} + \frac{\mu_0 I}{\pi a} = \frac{\mu_0 I}{2\pi a} + \frac{2\mu_0 I}{2\pi a} = \frac{3\mu_0 I}{2\pi a} \]

6. **Switching Off the Current in Wire 2**:
- When the current \( 2I \) in Wire 2 is switched off, the magnetic field due to Wire 2 becomes zero.
- Therefore, the net magnetic field at the midpoint will now only be due to Wire 1:
\[ B' = B_1 = \frac{\mu_0 I}{2\pi a} \]

7. **Conclusion**:
- The magnetic field at the midpoint after switching off the current \( 2I \) in Wire 2 will be:
\[ B' = \frac{\mu_0 I}{2\pi a} \]

### Final Answer:
The magnetic field at the midpoint after switching off the current \( 2I \) will be \( \frac{\mu_0 I}{2\pi a} \).