Two long parallel wires are separated by a distance of 2m. They carry a current of `1A` each in opposite direction. The magnetic induction at the midpoint of a straight line connecting these two wires is

Options

A. zero
B. `2xx10^(-7)T`
C. `4xx10^(-5)T`
D. `4xx10^(-7)T`

Explanation

To solve the problem of finding the magnetic induction at the midpoint of two long parallel wires carrying currents in opposite directions, we can follow these steps:

### Step 1: Understand the Setup
We have two long parallel wires separated by a distance of 2 meters, with each carrying a current of 1 A in opposite directions. The midpoint between the two wires is 1 meter away from each wire.

### Step 2: Use the Formula for 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 the formula:
\[ B = \frac{\mu_0 I}{2 \pi r} \]
where:
- \( \mu_0 \) is the permeability of free space (\( \mu_0 = 4\pi \times 10^{-7} \, \text{T m/A} \)),
- \( I \) is the current in amperes,
- \( r \) is the distance from the wire in meters.

### Step 3: Calculate the Magnetic Field from Each Wire at the Midpoint
Since the midpoint is 1 meter away from each wire, we can calculate the magnetic field due to each wire at this point.

For Wire 1 (carrying current \( I_1 = 1 \, \text{A} \)):
\[ B_1 = \frac{\mu_0 I_1}{2 \pi r_1} = \frac{4\pi \times 10^{-7} \times 1}{2 \pi \times 1} = \frac{4 \times 10^{-7}}{2} = 2 \times 10^{-7} \, \text{T} \]

For Wire 2 (carrying current \( I_2 = 1 \, \text{A} \) in the opposite direction):
\[ B_2 = \frac{\mu_0 I_2}{2 \pi r_2} = \frac{4\pi \times 10^{-7} \times 1}{2 \pi \times 1} = \frac{4 \times 10^{-7}}{2} = 2 \times 10^{-7} \, \text{T} \]

### Step 4: Determine the Direction of the Magnetic Fields
Using the right-hand rule:
- For Wire 1, the magnetic field at the midpoint will be directed out of the page.
- For Wire 2, the magnetic field at the midpoint will be directed into the page.

### Step 5: Calculate the Net Magnetic Field
Since the magnetic fields from the two wires are in opposite directions, we subtract the magnitudes:
\[ B_{\text{net}} = B_1 - B_2 = 2 \times 10^{-7} - 2 \times 10^{-7} = 0 \, \text{T} \]

### Conclusion
The magnetic induction at the midpoint of the straight line connecting the two wires is:
\[ B_{\text{net}} = 0 \, \text{T} \]