A long, straight wire carries a 13.0 A current. An electron...

A long, straight wire carries a 13.0 A current. An electron is fired parallel to this wire with a velocity of $250 \mathrm{~km} / \mathrm{s}$ in the same direction as the current, $2.00 \mathrm{~cm}$ from the wire. (a) Find the magnitude and direction of the electron's initial acceleration. (b) What should be the magnitude and direction of a uniform electric field that will allow the electron to continue to travel parallel to the wire? (c) Is it necessary to include the effects of gravity? Justify your answer.

Key Concepts

Magnetic Field from a Current-Carrying Wire

This concept involves the magnetic field generated by a current flowing through a conductor. The magnetic field around a long, straight wire can be determined using Ampère’s law or the Biot–Savart law, which shows that the field decreases with distance from the wire and circulates around it. This principle underpins the calculation of forces on moving charges in the vicinity of current-carrying conductors.

Lorentz Force

The Lorentz force is the force experienced by a charged particle moving in a magnetic field. It is defined by the equation F = q(v × B), where F is the force, q is the charge of the particle, v is its velocity, and B is the magnetic field. This concept is key to understanding how a moving electron is accelerated in the magnetic field generated by the current-carrying wire.

Newton’s Second Law

Newton’s Second Law relates the force acting on an object to its acceleration through the equation F = ma, where F is the net force, m is the mass of the object, and a is its acceleration. This principle is used to compute the acceleration of the electron once the magnetic force acting on it has been determined.

Electric Field Force and Force Cancellation

This concept involves using a uniform electric field to apply an electric force on a charged particle, described by F = qE. In situations where a moving charged particle experiences an undesired magnetic force, a suitably chosen electric field can be used to counteract the magnetic force, thus allowing the particle to continue along its initial path without deflection.

Negligible Gravitational Effects

In the context of charged particles moving under electromagnetic forces, gravitational forces are often insignificant due to the very small mass of electrons compared to the magnitude of electromagnetic forces. This concept is important for justifying the omission of gravity in many electromagnetism problems dealing with particles like electrons.

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