The drawing shows four insulated wires overlapping one another, forming a square with 0.050-m sides. All four wires are much longer than the sides of the square. The net magnetic field at the center of the square is 91.5 µT. Calculate the current I.
It matters how the current is flowing in each of the wires.
If the currents are flowing in the same direction, they repel one another and you take the difference of I for wires in parallel with each other. If they flow in the same direction you take the sum of I. Do that for both sides of parallel lines and add your equations then set the equation equal to the net magnetic field in the center.
To find the current I, we can use Ampere's law and the formula for the magnetic field produced by a long, straight wire.
Ampere's law states that the integral of the magnetic field around a closed loop is equal to the product of the current enclosed by the loop and the permeability of free space (μ₀). In equation form, it is expressed as:
∮ B · dl = μ₀ I_enclosed
Applying Ampere's law to our problem, we consider a square loop in the plane of the wires. The magnetic field at the center of the square is the sum of the magnetic fields produced by each wire. Since the wires are insulated and not connected, the current enclosed by the loop is the sum of the currents in each wire.
Now, let's calculate the magnetic field produced by a long, straight wire at the center of the square. The formula for the magnetic field produced by a long, straight wire at a distance r from the wire is:
B = (μ₀ I) / (2πr)
Considering that the length of the wire is much longer than the sides of the square, we can approximate it as a straight wire.
Let's denote the current in each wire as I. Since all four wires overlap and form a square, the magnetic field at the center of the square is the sum of the magnetic fields produced by each wire:
B_total = 4 * B
Substituting the formula for B, we have:
B_total = 4 * (μ₀ I) / (2πr)
Given B_total = 91.5 µT and r = 0.050 m, we can rearrange the equation to solve for the current I:
I = (B_total * 2πr) / (4μ₀)
Now, plug in the values into the equation and calculate:
I = (91.5 µT * 2π * 0.050 m) / (4 * π * 10^(-7) T·m/A)
I = 9.15 * 10^(-5) A
Therefore, the current I is approximately 9.15 * 10^(-5) A.