And i2 being in the same direction. The wires are separated by a distance d. What is the magnetic field at a point midway between the wires?

To determine the magnetic field at a point midway between the wires, we can use the Biot-Savart Law.

The Biot-Savart Law states that the magnetic field produced by a current-carrying wire at a point is directly proportional to the current in the wire and inversely proportional to the distance from the wire.

Let's assume that the wire carrying current i1 is located at x = -d/2, and the wire carrying current i2 is located at x = d/2.

The magnetic field produced by the wire carrying current i1 at the point midway between the wires (x = 0) can be calculated using the Biot-Savart Law:

dB1 = (μ₀/4π) * (i1 * dl1 × r1) / r1^3

where dB1 is the magnetic field produced by the wire carrying current i1, μ₀ is the permeability of free space, i1 is the current in the wire carrying i1, dl1 is a small element of the wire carrying i1, r1 is the distance from the small element dl1 to the point midway between the wires.

Similarly, the magnetic field produced by the wire carrying current i2 at the point midway between the wires (x = 0) can be calculated using the Biot-Savart Law:

dB2 = (μ₀/4π) * (i2 * dl2 × r2) / r2^3

where dB2 is the magnetic field produced by the wire carrying current i2, μ₀ is the permeability of free space, i2 is the current in the wire carrying i2, dl2 is a small element of the wire carrying i2, r2 is the distance from the small element dl2 to the point midway between the wires.

Since the wires are parallel and the currents are in the same direction, the magnetic fields produced by each wire will have the same direction. Therefore, we can add the magnetic fields produced by each wire to find the total magnetic field at the point midway between the wires:

B = dB1 + dB2

To simplify the calculation, we can assume that the wires are infinitely long and straight, so that the magnetic fields produced by each wire are uniform over the entire length of the wires.

Since the wires are parallel, the distance from each small element dl1 (or dl2) to the point midway between the wires is the same. Let's call this distance r.

Therefore, r1 = r2 = r.

The magnetic field produced by each wire at the point midway between the wires can be calculated as:

dB1 = (μ₀/4π) * (i1 * dl1 × r) / r^3
dB2 = (μ₀/4π) * (i2 * dl2 × r) / r^3

Since the wires are parallel and the currents are in the same direction, the dl1 and dl2 vectors will have the same magnitude and direction.

Therefore, we can rewrite the equations as:

dB1 = (μ₀/4π) * (i1 * dl × r) / r^3
dB2 = (μ₀/4π) * (i2 * dl × r) /