An electron moving in a direction perpendicular to a uniform...

An electron moving in a direction perpendicular to a uniform magnetic field at a speed of $1.5 \times 10^{7} \mathrm{~m} / \mathrm{s}$ undergoes an acceleration of $4.0 \times 10^{16} \mathrm{~m} / \mathrm{s}^{2}$ to the right (the positive $x$ -direction) when its velocity is upward (the positive $y$ -direction). Determine the magnitude and direction of the field.

Key Concepts

Lorentz Force

The Lorentz force is the fundamental concept governing the behavior of charged particles in electromagnetic fields. It states that a moving charge in a magnetic field experiences a force given by F = q(v × B). This force is always perpendicular to both the velocity of the charge and the magnetic field, which means it can alter the direction of the particle’s motion without doing work on it.

Cross Product and Direction Determination

The force experienced by a charged particle due to a magnetic field is determined by the cross product of the velocity and the magnetic field vectors. Understanding the vector nature of this product, and applying the right-hand rule (or its modified form for negative charges like electrons), allows one to deduce the direction of the force acting on the particle, which is essential for predicting the particle’s subsequent motion.

Uniform Magnetic Field

A uniform magnetic field is one where the field strength and direction are constant in space. This uniformity simplifies the analysis of moving charges since the magnetic force remains consistent throughout the particle’s motion. In problems involving uniform magnetic fields, the resulting motion of charged particles is often circular or helical, with the magnetic force continuously providing the necessary centripetal acceleration.

Centripetal Acceleration in Charged Particle Motion

When a charged particle moves perpendicular to a uniform magnetic field, the force acting on it continually changes its direction, causing circular motion. The magnitude of this centripetal acceleration is linked to the speed of the particle and the magnetic field strength. By equating the magnetic force (from the Lorentz equation) to the centripetal force required for circular motion, one can solve for unknown quantities such as the magnetic field strength.

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