Two long parallel wires are placed at a distance of `12cm` from each other in air. Each wire has a current of `3A`. Calculate the magnetic field at a point mid way between them when the currents in them are : (i) in the same direction and (ii) in opposite direction.

Explanation

To solve the problem of calculating the magnetic field at a point midway between two long parallel wires carrying currents, we will follow these steps:

### Given Data:
- Distance between the wires, \( d = 12 \, \text{cm} = 0.12 \, \text{m} \)
- Current in each wire, \( I = 3 \, \text{A} \)

### Step 1: Calculate the distance from each wire to the midpoint
Since the wires are parallel and the point of interest is midway between them, the distance from each wire to the midpoint is:
\[ D = \frac{d}{2} = \frac{0.12}{2} = 0.06 \, \text{m} \]

### Step 2: Calculate the magnetic field due to one wire
The magnetic field \( B \) due to a long straight wire at a distance \( D \) is given by the formula:
\[ B = \frac{\mu_0 I}{2 \pi D} \]
where \( \mu_0 \) (the permeability of free space) is approximately \( 4\pi \times 10^{-7} \, \text{T m/A} \).

Substituting the values:
\[ B = \frac{(4\pi \times 10^{-7}) \times 3}{2 \pi \times 0.06} \]
\[ B = \frac{(4 \times 3 \times 10^{-7})}{2 \times 0.06} \]
\[ B = \frac{12 \times 10^{-7}}{0.12} = 10^{-6} \, \text{T} = 1 \, \mu\text{T} \]

### Step 3: Case (i) - Currents in the same direction
When the currents are in the same direction, the magnetic fields due to both wires at the midpoint will be in the same direction (let's say into the page).

Thus, the total magnetic field \( B_{\text{total}} \) at the midpoint is:
\[ B_{\text{total}} = B_1 + B_2 = 1 \, \mu\text{T} + 1 \, \mu\text{T} = 2 \, \mu\text{T} \]

### Step 4: Case (ii) - Currents in opposite directions
When the currents are in opposite directions, the magnetic fields due to the wires at the midpoint will be in opposite directions.

Thus, the total magnetic field \( B_{\text{total}} \) at the midpoint is:
\[ B_{\text{total}} = B_1 - B_2 = 1 \, \mu\text{T} - 1 \, \mu\text{T} = 0 \, \text{T} \]

### Final Answers:
(i) When currents are in the same direction:
\[ B_{\text{total}} = 2 \, \mu\text{T} \]

(ii) When currents are in opposite directions:
\[ B_{\text{total}} = 0 \, \text{T} \]