A straight wire is aligned east-west in a region where...

A straight wire is aligned east-west in a region where Earth's magnetic field has magnitude $0.048 \mathrm{mT}$ and direction $72^{\circ}$ below the horizontal, with the horizontal component directed due north. The wire carries a current $I$ toward the west. The magnetic force on the wire per unit length of wire has magnitude $0.020 \mathrm{N} / \mathrm{m}$. (a) What is the direction of the magnetic force on the wire? (b) What is the current $I ?$

A straight wire is aligned east-west in a region where Earth's magnetic field has magnitude $0.048 \mathrm{mT}$ and direction $72^{\circ}$ below the horizontal, with the horizontal component directed due north. The wire carries a current $I$ toward the west. The magnetic force on the wire per unit length of wire has magnitude $0.020 \mathrm{N} / \mathrm{m}$.
(a) What is the direction of the magnetic force on the wire? (b) What is the current $I ?$

Key Concepts

Cross Product and the Right-Hand Rule

The cross product operation is used to calculate both the magnitude and the direction of the force experienced by the conductor, where the magnitude is given by |F| = I L B sin(?) and the direction is determined using the right-hand rule. This rule states that if the thumb represents the current direction and the fingers represent the magnetic field, then the force points in the direction of the palm, which is crucial for predicting the force's orientation.

Decomposition of Magnetic Field Vectors

In many practical situations, the magnetic field may have both horizontal and vertical components. Understanding how to decompose the Earth's magnetic field into these components allows the correct application of the magnetic force equation. The angle by which the field is inclined below the horizontal is used to calculate the contributions to the force, especially when the wire orientation is not aligned with one of the cardinal directions.

Magnetic Force on a Current-Carrying Conductor

This concept involves the principle that a current-carrying conductor in a magnetic field experiences a force given by the equation F = I L × B. This force is a result of the interaction between the electric current (due to moving charges) and the magnetic field, and its direction is determined by the cross product of the current direction and the magnetic field vector.

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