. 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. (b) Each current is doubled, so that I1 becomes 10.0 A and I2 becomes 4.00 A. Now what is the magnitude of the force that each wire exerts on a 1.20-m length of the other?

Verified step by step guidance

1. Identify the direction of the currents in the two wires. Since the currents are in opposite directions, the magnetic force between the wires will be repulsive according to the right-hand rule.
2. Use the formula for the magnetic force per unit length between two parallel currents: \( F/L = \frac{\mu_0 \cdot I_1 \cdot I_2}{2\pi \cdot d} \), where \( \mu_0 \) is the permeability of free space (\( 4\pi \times 10^{-7} \, T\cdot m/A \)), \( I_1 \) and \( I_2 \) are the currents in the wires, and \( d \) is the distance between the wires.
3. Substitute the given values into the formula: \( I_1 = 10.0 \, A \), \( I_2 = 4.00 \, A \), and \( d = 0.400 \, m \).
4. Calculate the force per unit length using the values substituted in the formula.
5. To find the total force exerted on a 1.20-m length of each wire, multiply the force per unit length by the length of the wire segment: \( F = (F/L) \cdot 1.20 \, m \).

Key Concepts

Magnetic Force Between Parallel Currents

When two parallel wires carry electric currents, they exert a magnetic force on each other. The direction of this force depends on the direction of the currents: if the currents flow in the same direction, the wires attract each other; if they flow in opposite directions, they repel. The magnitude of this force can be calculated using the formula F = (μ₀/2π) * (I₁ * I₂ * L / d), where μ₀ is the permeability of free space, I₁ and I₂ are the currents, L is the length of the wire segment, and d is the distance between the wires.

Ampere's Law

Ampere's Law relates the integrated magnetic field around a closed loop to the electric current passing through the loop. It is fundamental in understanding how currents create magnetic fields and how these fields interact with other currents. In the context of parallel wires, it helps derive the force between them by considering the magnetic field produced by one wire at the location of the other wire.

Superposition Principle

The superposition principle states that when multiple forces act on an object, the total force is the vector sum of the individual forces. In the case of the two wires, each wire experiences a force due to the current in the other wire. By applying the superposition principle, we can calculate the net force on each wire by considering the contributions from both currents, allowing us to analyze the system comprehensively.