A wire carries a steady current of 2.40 A. A straight...

A wire carries a steady current of 2.40 A. A straight section of the wire is $0.750 \mathrm{m}$ long and lies along the $x$ axis within uniform magnetic field, $\overrightarrow{\mathbf{B}}=160 \hat{\mathbf{k}}$ T. If the current is in the positive $x$ direction, what is the magnetic force on the section of wire?

Key Concepts

Magnetic Force on a Current-Carrying Conductor

This concept states that a conductor carrying an electric current experiences a force when it is placed in a magnetic field. The magnitude and direction of this force depend on the current in the wire, the length of the wire segment within the magnetic field, and the orientation of the wire relative to the field. This principle is central to understanding the interaction between electricity and magnetism in devices like motors and generators.

Lorentz Force Law

The Lorentz force law describes the force exerted on a charged particle moving through a magnetic field, and it extends to a current-carrying wire by considering the collective effect of moving charges. In the context of a wire, the force is proportional to the current, the length of the conductor in the field, and the magnetic field strength, while also depending on the angle between the current's direction and the magnetic field.

Vector Cross Product

The vector cross product is a mathematical operation that takes two vectors and produces a third vector, which is perpendicular to the plane containing the first two. In the formula for the magnetic force on a current-carrying conductor, the use of the cross product determines not only the magnitude of the force but also its direction relative to both the current and the magnetic field.

Right-Hand Rule

The right-hand rule is a mnemonic used to determine the direction of the resultant vector from a cross product. When applying this rule to the magnetic force on a current-carrying conductor, one points the fingers in the direction of the current, curls them toward the direction of the magnetic field, and the thumb then points in the direction of the force acting on the conductor.

Uniform Magnetic Field

A uniform magnetic field is one where the magnetic field strength and direction are constant throughout a region of space. This uniformity simplifies the calculation of the force on a current-carrying wire since the magnetic field does not change along the path of the wire segment, allowing direct application of the force formula without additional spatial complications.

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