. Two long, parallel wires are separated by a distance of 0.400 m (Fig. E28.29). The currents I1 and I2 have the directions shown. (a) Calculate the magnitude of the force exerted by each wire on a 1.20-m length of the other. Is the force attractive or repulsive?

Verified step by step guidance

1. Identify the direction of the currents in both wires. If the currents are in the same direction, the force between the wires will be attractive. If the currents are in opposite directions, the force will be repulsive.
2. Use Ampere's Law to determine the magnetic field produced by one wire at the location of the other. The magnetic field, B, due to a long straight wire carrying a current I at a distance r is given by the formula B = \(\frac{\mu_0 I}{2\pi r}\), where \(\mu_0\) is the permeability of free space.
3. Calculate the force on a segment of the second wire due to the magnetic field produced by the first wire. The force per unit length, f, on a wire carrying a current I in a magnetic field B is given by f = I B. Use the length of the wire segment given in the problem to find the total force.
4. Substitute the values for the currents, the distance between the wires, and the length of the wire segment into the formulas to find the magnetic field and the force.
5. Determine the nature of the force (attractive or repulsive) based on the direction of the currents and summarize the findings.

Key Concepts

Magnetic Field Due to a Current-Carrying Wire

A current-carrying wire generates a magnetic field around it, which can be described by the right-hand rule. The strength of this magnetic field at a distance 'r' from the wire is given by the formula B = (μ₀I)/(2πr), where μ₀ is the permeability of free space and I is the current. This magnetic field interacts with other currents, leading to forces between parallel wires.

Force Between Parallel Current-Carrying Wires

When two parallel wires carry currents, they exert forces on each other due to their magnetic fields. The force per unit length between the wires can be calculated using the formula F/L = (μ₀I₁I₂)/(2πd), where F is the force, L is the length of the wire, I₁ and I₂ are the currents, and d is the distance between the wires. The direction of the force is determined by the direction of the currents: if they flow in the same direction, the force is attractive; if in opposite directions, it is repulsive.

Right-Hand Rule

The right-hand rule is a mnemonic used to determine the direction of the magnetic field and the force on a current-carrying conductor. For a straight wire, if you point your thumb in the direction of the current, your fingers curl in the direction of the magnetic field lines. This rule helps visualize the interactions between currents and magnetic fields, essential for predicting the behavior of the wires in the problem.