Two long, straight wires are parallel and 10 cm apart. One carries a current of 2.0 A, the other a current of 5.0 A. (a) If the two currents flow in opposite directions, what is the magnitude and direction of the force per unit length of one wire on the other? (b) What is the magnitude and direction of the force per unit length if the currents flow in the same direction?
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(a) When the currents flow in opposite directions, the force per unit length between the wires can be calculated using Ampere's Law. According to Ampere's Law, the magnetic field produced by a long, straight wire carrying current I at a distance r from the wire is given by B = μ₀I / (2πr), where μ₀ is the magnetic constant.
The force per unit length between the wires is given by the formula F = ILB, where I is the current in one wire, L is the length of the wire, and B is the magnetic field produced by the other wire.
In this case, the current in one wire is 2.0 A, and the current in the other wire is 5.0 A. The distance between the wires is 10 cm, which is equal to 0.1 m.
The force per unit length on one wire caused by the magnetic field produced by the other wire can be calculated as follows:
F = (2.0 A) * (0.1 m) * (μ₀ * 5.0 A) / (2π * 0.1 m)
= μ₀ * (2.0 A) * (5.0 A) / (2π)
= (4π × 10^(-7) T·m/A) * (2.0 A) * (5.0 A) / (2π)
= (4π × 10^(-7) T·m/A) * (10 A²) / (2π)
= (40 × 10^(-7) T·m²/A)
= 4 × 10^(-6) N/m
Therefore, the magnitude of the force per unit length of one wire on the other when the currents flow in opposite directions is 4 × 10^(-6) N/m.
The direction of the force can be determined by the right-hand rule. Point your right thumb in the direction of the current in the first wire. Your fingers will curl in the direction of the magnetic field produced by the second wire. The force per unit length between the wires is directed in the direction that your curled fingers point, which is perpendicular to the plane formed by the two wires.
(b) When the currents flow in the same direction, the force per unit length between the wires is attractive, and its magnitude can be calculated using the same formula as before.
F = (2.0 A) * (0.1 m) * (μ₀ * 2.0 A) / (2π * 0.1 m)
= (4π × 10^(-7) T·m/A) * (2.0 A) * (2.0 A) / (2π)
= (40 × 10^(-7) T·m²/A)
= 4 × 10^(-6) N/m
Therefore, the magnitude of the force per unit length of one wire on the other when the currents flow in the same direction is 4 × 10^(-6) N/m.
The direction of the force is attractive, causing the wires to pull towards each other.
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