Two long, parallel wires, each with a mass per unit length of 40 g / m, are supported in a hori

step 1 This problem covers the concept of the magnetic force between the two current carrying wire. And

Two long, parallel wires, each with a mass per unit length...

Two long, parallel wires, each with a mass per unit length of $40 \mathrm{~g} / \mathrm{m}$, are supported in a horizontal plane by $6.0-\mathrm{cm}-$ long strings, as shown in Figure P19.72. Each wire carries the same current $I$, causing the wires to repel each other so that the angle $\theta$ between the supporting strings is $16^{\circ}$. (a) Are the currents in the same or opposite directions? (b) Determine the magnitude of each current.

Two long, parallel wires, each with a mass per unit length of $40 \mathrm{~g} / \mathrm{m}$, are supported in a horizontal plane by $6.0-\mathrm{cm}-$ long strings, as shown in Figure P19.72. Each wire carries the same current $I$, causing the wires to repel each other so that the angle $\theta$ between the supporting strings is $16^{\circ}$.
(a) Are the currents in the same or opposite directions?
(b) Determine the magnitude of each current.

Key Concepts

Magnetic Interaction Between Currents

Parallel current-carrying wires interact through magnetic forces. When the currents flow in the same direction, the wires attract each other; when they flow in opposite directions, the wires repel. This fundamental behavior is described by Ampère’s force law, where the force per unit length between two long, straight wires is proportional to the product of the currents and inversely proportional to the distance between them.

Mechanical Equilibrium in Static Systems

In a static system, all forces must balance so that the net force on each component is zero. For suspended objects, this means that the vector sum of forces, including gravitational forces, tension from support strings, and any additional forces (like magnetic forces), must cancel out. This principle allows one to set up equations along the vertical and horizontal directions to solve for unknown quantities in the system.

Vector Decomposition and Free-Body Analysis

A free-body diagram is a critical tool in analyzing physical systems by visually representing all forces acting on an object. Vector decomposition involves breaking forces into perpendicular components—typically horizontal and vertical—which simplifies the process of applying equilibrium conditions. This method is essential in determining relationships between forces such as tension and magnetic repulsion, especially when angles (like the angle of the supporting strings) are involved.

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