Two parallel long straight wires are suspended by strings of...

Two parallel long straight wires are suspended by strings of length $L=1.2 \mathrm{~m} .$ Each wire has mass per unit length $\quad 0.050 \quad \mathrm{~kg} / \mathrm{m}$. When the wires each carry $50.0 \mathrm{~A}$ of current, the wires swing apart. (a) How far apart are the wires in equilibrium? (Assume that this distance is small compared to $L$.) [Hint: Use a small angle approximation.] (b) Are the wires carrying current in the same or opposite directions?

Key Concepts

Current Direction and Magnetic Interaction

The direction of current flow in wires determines the nature of the magnetic interaction between them. If the currents are in the same direction, the wires attract each other due to the direction of the magnetic fields created around them. Conversely, if the currents are in opposite directions, the magnetic fields interact to produce a repulsive force. This principle is critical when interpreting observations such as wires swinging apart, which implies repulsion and thus indicates that the currents are in opposite directions.

Small Angle Approximation

The small angle approximation is a useful technique in physics to simplify trigonometric relationships when the angles involved are small (measured in radians). Under this approximation, sin(?) is approximated by ? and cos(?) by 1. This simplification turns a potentially complex trigonometric problem into a linear one, making it easier to relate the displacement of the pendulum to the forces acting on it.

Force Balance in Equilibrium

For objects in equilibrium, all forces must balance such that the net force is zero. In the context of a suspended wire experiencing a horizontal magnetic repulsion and gravity, this means that the horizontal magnetic force is exactly balanced by the horizontal component of the tension in the string. Analyzing this balance allows one to derive relationships among the magnetic force, the displacement of the wires, and the physical parameters of the setup (like mass per unit length, string length, and gravitational acceleration).

Magnetic Force Between Parallel Currents

When currents flow through parallel wires, they interact magnetically. According to Ampère’s force law, two wires carrying currents experience a force per unit length that is proportional to the product of the currents and inversely proportional to the distance between them. Importantly, if the currents flow in the same direction, the wires attract; if they flow in opposite directions, they repel. In problems where the wires swing apart, it indicates a repulsive force that results from opposing current directions.

Pendulum Equilibrium

In situations where wires are suspended by strings, each wire behaves much like a pendulum. The forces acting on the wires include both the magnetic force (which might be horizontal) and gravity (vertical). At equilibrium, the wires settle at an angle where the horizontal magnetic force is balanced by the horizontal component of the tension in the string, while the vertical forces (gravity and the vertical component of tension) balance out. This equilibrium analysis is key to linking the observed displacement with the forces involved.

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