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. 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 formula for the magnetic force between two parallel currents: The force per unit length between two parallel wires carrying currents I1 and I2 is given by the formula: $F=\frac{\mu oI}{}$₁I₂2πd, where $\mu o$ is the permeability of free space, $d$ is the distance between the wires, and $I$₁ and $I$₂ are the currents in the wires.
2. Substitute the given values into the formula: Use the given distance $d=0.400\text{m}$ and the length of the wire $L=1.20\text{m}$. The formula for the force on a length $L$ of wire becomes: $F=\frac{\mu oI}{}$₁I₂L2πd.
3. Consider the direction of the currents: The direction of the currents determines whether the force is attractive or repulsive. If the currents are in the same direction, the force is attractive. If the currents are in opposite directions, the force is repulsive.
4. Calculate the force using the formula: Plug in the values for $\mu o$ (which is $4ϗ10-7\text{T m/A}$), $I$₁, $I$₂, $L$, and $d$ into the formula to find the magnitude of the force.
5. Interpret the result: Based on the calculation and the direction of the currents, determine whether the force is attractive or repulsive. This will help understand the interaction between the two wires.

Key Concepts

Magnetic Force Between Parallel Currents

When two parallel wires carry currents, they exert magnetic forces on each other. The direction of the force depends on the direction of the currents: if the currents are in the same direction, the force is attractive; if opposite, the force is repulsive. The magnitude of the force per unit length is given by the formula F/L = (μ₀/2π) * (I₁I₂/d), where μ₀ is the permeability of free space, I₁ and I₂ are the currents, and d is the distance between the wires.

Permeability of Free Space

The permeability of free space, denoted as μ₀, is a fundamental physical constant that describes the ability of a vacuum to support the formation of a magnetic field. Its value is approximately 4π × 10⁻⁷ T·m/A. This constant is crucial in calculating the magnetic forces between currents and is used in the formula for the force between parallel wires.

Direction of Magnetic Forces

The direction of the magnetic force between two current-carrying wires is determined by the right-hand rule. For parallel wires, if the currents flow in the same direction, the wires attract each other; if the currents flow in opposite directions, the wires repel each other. This concept is essential for determining whether the force in the given problem is attractive or repulsive.