Three long, straight, parallel wires each carry a current of 10 A in the positive x direction. If the distance between each wire and the other two is 10 cm, what is the magnitude of the magnetic force on a 20-cm length of either of the wires?
A) 57 μN
B) 40 μN
C) 69 μN
D) 50 μN
E) 20 μN

Question

Quizplus · Accepted Answer

The magnetic force on a wire is given by:
F = ILB
where I is the current in the wire, L is the length of the wire, and B is the magnetic field.

The magnetic field at a distance r from a long, straight, current-carrying wire is given by:
B = μ0I/2πr
where μ0 is the permeability of free space.

For the wires in this problem, the magnetic field at any point on one wire due to the other two wires is equal in magnitude and opposite in direction to the field due to the third wire. Therefore, the net magnetic force on a length of wire is zero.

However, if we consider a section of wire that is not centered between the other two wires, the magnetic field due to the other two wires will not be exactly opposite to the field due to the third wire, and there will be a net magnetic force on that section of wire.

Let's consider a section of wire at a distance x from one of the other wires. The magnetic field due to the other two wires at that point is:
B1 = μ0I/(2π(0.1-x))
B2 = μ0I/(2π(0.1+x))
where x is measured from the center of the other wire.

The magnetic field due to the third wire is:
B3 = μ0I/(2π(0.1))
since it is at the same distance from the section of wire.

The net magnetic field is:
Bnet = B3 - (B1 + B2)
Bnet = μ0I/(2π0.1) - μ0I/(2π(0.1-x)) - μ0I/(2π(0.1+x))
Bnet = (2μ0IxI)/(4πx(x^2-0.01))

The direction of the magnetic force can be determined using the right-hand rule. If we assume that the section of wire has current in the positive x direction, then the force will be in the positive y direction.

The magnitude of the magnetic force is:
F = ILBnet = (10A)(0.2m)Bnet
F = (2μ0Ix)/(4π(x^2-0.01))

We can calculate this for x = 0.05 m, which gives:
F = 6.869 μN

Since there are two sections of wire (one on each side of the wire), the total magnetic force on the wire is twice this amount, or:
Ftotal = 2F = 13.74 μN

Therefore, the best choice is C.

Three Long, Straight, Parallel Wires Each Carry a Current of 10