A long straight wire carries a current of 50 A. An electron,...

A long straight wire carries a current of 50 A. An electron, traveling at $1.0 \times 10^{7} \mathrm{m} / \mathrm{s}$ , is 5.0 $\mathrm{cm}$ from the wire. What is the magnitude of the magnetic force on the electron if the electron velocity is directed (a) toward the wire, (b) parallel to the wire in the direction of the current, and (c) perpendicular to the two directions defined by (a) and (b)?

A long straight wire carries a current of 50 A. An electron, traveling at $1.0 \times 10^{7} \mathrm{m} / \mathrm{s}$ , is 5.0 $\mathrm{cm}$ from the wire. What is the magnitude of the magnetic force on the electron if the electron velocity
is directed (a) toward the wire, (b) parallel to the wire in the direction of the current, and (c) perpendicular to the two directions defined by (a) and (b)?

Key Concepts

Magnetic Field of a Current-Carrying Wire

A long straight current-carrying conductor generates a magnetic field that circles the wire. Its magnitude can be calculated by the expression B = (??I)/(2?r), where I is the current and r is the distance from the wire. This formula illustrates how the field decays with distance and is foundational to understanding how electrical currents produce magnetic effects.

Lorentz Force

The magnetic force experienced by a moving charged particle in a magnetic field is given by F = q(v × B). This force, known as the Lorentz force, depends on the charge of the particle, its velocity, and the magnetic field it traverses. The cross product indicates that the force is perpendicular to both the velocity and the magnetic field, which determines its direction and magnitude.

Cross Product and Directional Dependence

The cross product in the Lorentz force equation means that only the component of the velocity perpendicular to the magnetic field contributes to the force. If the velocity is parallel to the magnetic field, the force is zero. This directional dependence is crucial for analyzing situations where the orientation of the velocity vector relative to the magnetic field changes.

Right-Hand Rule

The right-hand rule is a mnemonic used to determine the direction of the magnetic field around a current or the force on a charged particle. For a current-carrying wire, if you point your thumb in the direction of the current, your curled fingers show the direction of the magnetic field. For the Lorentz force, pointing your fingers in the direction of velocity and curling them toward the magnetic field shows the direction of the force for a positive charge; for an electron, which is negatively charged, the force is opposite.

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