Four long, straight wires are parallel to each other and their cross-section forms a square. Each side of the square measures 0.0112 m. If each wire carries a currrent of 8.33 A in the direction shown, find the magnitude of the total magnetic field at P, the center of the square.

Find the B due to one wire, and add or subtract the B (same magnitude) in accordance with the right hand rule.

To find the magnitude of the total magnetic field at point P, the center of the square, we can first calculate the magnetic field generated by a single wire at P, and then sum up the contributions from all four wires.

The magnetic field generated by a long straight wire at a point P, a distance 'r' away from the wire, is given by the formula:

B = (μ₀ * I) / (2 * π * r)

Where:
B is the magnetic field,
μ₀ is the permeability of free space (4π × 10^(-7) T•m/A),
I is the current flowing through the wire, and
r is the distance from the wire to the point 'P'.

Since each wire is carrying a current of 8.33 A, the magnetic field generated by a single wire at the center point 'P', located equidistant from all four wires, would be:

B = (4π × 10^(-7) T•m/A * 8.33 A) / (2 * π * (0.0112/2) m)

Simplifying further:

B = (4π × 10^(-7) * 8.33) / 0.0112 T

Now, we need to multiply this value by 4 since there are four wires in total:

Total magnetic field at P = 4 * B

Total magnetic field at P = 4 * (4π × 10^(-7) * 8.33) / 0.0112 T

Calculating this value will give us the magnitude of the total magnetic field at point P.

To find the magnitude of the total magnetic field at point P, we can use the Biot-Savart Law, which relates the magnetic field produced by a current-carrying wire to the distance from the wire.

The Biot-Savart Law states that the differential magnetic field produced at a point by a small length segment of a wire carrying a current is given by:

dB = (μ₀/4π) * (I * dl × r̂) / r²

where:
- dB is the differential magnetic field
- μ₀ is the permeability of free space (4π × 10^-7 T⋅m/A)
- I is the current
- dl is an infinitesimal length element of the wire
- r̂ is the unit vector in the direction of r (the vector pointing from the wire element to the point where the magnetic field is being calculated)
- r is the distance from the wire element to the point where the magnetic field is being calculated.

Since the wires are parallel and form a square, the direction of the magnetic field produced by each wire at point P will cancel out for the opposite sides. This means that the only magnetic fields that contribute to the total magnetic field at P are the ones produced by the two adjacent wires.

Let's label the wires as A, B, C, and D, and assume that the current in wires A and C flow upwards, while the current in wires B and D flow downwards. Now, we will calculate the magnetic field produced by wires A and C at point P.

The distance from wire A to point P is the distance from the center of the square to any of its sides: d = 0.0112 m / 2 = 0.0056 m.

Using the Biot-Savart Law, the magnitude of the magnetic field produced by wire A at point P is given by:

B₁ = (μ₀/4π) * (I * dl × r̂) / r²

To simplify this equation, let's assume that dl = L, the length of wire A, since the length is the same for all wires.

The magnetic field produced by wire C at point P will have the same magnitude as B₁, but will point in the opposite direction. Therefore, the total magnetic field at point P due to wires A and C will be zero.

Now, let's calculate the magnetic field produced by wires B and D at point P.

The distance from wire B to point P is also d = 0.0056 m.

Using the Biot-Savart Law, the magnitude of the magnetic field produced by wire B at point P is given by:

B₂ = (μ₀/4π) * (I * dl × r̂) / r²

Again, assuming dl = L, the length of wire B.

The magnetic field produced by wire D at point P will have the same magnitude as B₂, but will point in the opposite direction. Therefore, the total magnetic field at point P due to wires B and D will be zero.

Since the total magnetic field at point P is the sum of the magnetic fields due to wires A, B, C, and D, the magnitude of the total magnetic field at point P is 0 T.

Therefore, the magnitude of the total magnetic field at point P, the center of the square, is 0 T.