Two long, parallel wires are attracted to each other by a...

Two long, parallel wires are attracted to each other by a force per unit length of 320$\mu \mathrm{N} / \mathrm{m}$ . One wire carries a current of 20.0 $\mathrm{A}$ to the right and is located along the line $y=$ 0.500 $\mathrm{m}$ . The second wire lies along the $x$ axis. Determine the value of $y$ for the line in the plane of the two wires along which the total magnetic field is zero.

Key Concepts

Equating Magnetic Fields for Cancellation

In problems requiring the determination of locations where the net magnetic field is zero, one sets the magnitudes of the magnetic fields from different sources equal, accounting for their directions. This involves solving for positions where the contributions from each wire cancel out, a technique that combines the formula for magnetic fields with the superposition principle.

Ampère’s Force Law for Parallel Currents

Ampère’s force law describes the force per unit length between two parallel current-carrying wires, showing that the force is proportional to the product of the currents and inversely proportional to the distance between them. This principle is not only fundamental in calculating forces in current-carrying systems but also reinforces the understanding of how magnetic fields interact with one another.

Right-Hand Rule

The right-hand rule is a method used to determine the direction of the magnetic field around a current-carrying conductor. By pointing the thumb in the direction of the current, the curled fingers indicate the direction in which the magnetic field circles the wire. This rule is essential for correctly assessing the vector directions of the magnetic fields from different wires.

Superposition Principle

The superposition principle states that the total magnetic field at any point is the vector sum of the magnetic fields produced by each individual source. This allows for the analysis of multiple current-carrying wires by considering their contributions independently and then algebraically adding them to find conditions where, for example, the net field becomes zero.

Magnetic Field of a Long Straight Conductor

A long straight current-carrying wire produces a magnetic field whose magnitude is given by the formula B = ??I/(2?r), where I is the current, r is the distance from the wire, and ?? is the permeability of free space. This expression shows that the strength of the magnetic field diminishes with distance, making it a key factor in determining the field at various points around the wire.

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