Two long straight wires, each carrying a current `I` in opposite directions are separated by a distance `R`. The magnetic induction at a point mid way between the wire is

Options

A. zero
B. `(mu_0I)/(piR)`
C. `(2mu_I)/(piR)`
D. `(mu_0I)/(4piR)`

Explanation

To find the magnetic induction at a point midway between two long straight wires carrying currents in opposite directions, we can follow these steps:

### Step 1: Understand the Setup
We have two long straight wires, Wire 1 and Wire 2, each carrying a current `I` in opposite directions. The distance between the two wires is `R`, and we are interested in the magnetic field at a point `P` that is located midway between the two wires.

### Step 2: Determine the Distance from Each Wire
Since point `P` is midway between the two wires, the distance from each wire to point `P` is:
\[ d = \frac{R}{2} \]

### Step 3: Use the Formula for Magnetic Field
The magnetic field `B` due to a long straight wire carrying current `I` at a distance `d` from the wire is given by the formula:
\[ B = \frac{\mu_0 I}{2 \pi d} \]
where `μ₀` is the permeability of free space.

### Step 4: Calculate the Magnetic Field from Each Wire
For Wire 1, the magnetic field at point `P` (due to current `I` flowing in one direction) is:
\[ B_1 = \frac{\mu_0 I}{2 \pi \left(\frac{R}{2}\right)} = \frac{\mu_0 I}{\pi R} \]
For Wire 2, the magnetic field at point `P` (due to current `I` flowing in the opposite direction) is:
\[ B_2 = \frac{\mu_0 I}{2 \pi \left(\frac{R}{2}\right)} = \frac{\mu_0 I}{\pi R} \]

### Step 5: Determine the Direction of the Magnetic Fields
Since the currents are in opposite directions, the magnetic fields at point `P` due to each wire will be in the same direction (into the plane of the paper). Therefore, we can add the magnitudes of the magnetic fields from both wires.

### Step 6: Calculate the Net Magnetic Field
The net magnetic field `B_net` at point `P` is:
\[ B_{net} = B_1 + B_2 = \frac{\mu_0 I}{\pi R} + \frac{\mu_0 I}{\pi R} = \frac{2 \mu_0 I}{\pi R} \]

### Conclusion
Thus, the magnetic induction at the point midway between the two wires is:
\[ B_{net} = \frac{2 \mu_0 I}{\pi R} \]