An electron that is moving through a uniform magnetic field...

An electron that is moving through a uniform magnetic field has velocity $\vec{v}=(40 \mathrm{~km} / \mathrm{s}) \hat{\mathrm{i}}+(35 \mathrm{~km} / \mathrm{s}) \hat{\mathrm{j}}$ when it experiences a force $\vec{F}=-(4.2 \mathrm{fN}) \mathrm{i}+(4.8 \mathrm{fN}) \hat{\mathrm{j}}$ due to the magnetic field. If $B_{x}=0$, calculate the magnetic field $\vec{B}$.

An electron that is moving through a uniform magnetic field has velocity $\vec{v}=(40 \mathrm{~km} / \mathrm{s}) \hat{\mathrm{i}}+(35 \mathrm{~km} / \mathrm{s}) \hat{\mathrm{j}}$ when it experiences
a force $\vec{F}=-(4.2 \mathrm{fN}) \mathrm{i}+(4.8 \mathrm{fN}) \hat{\mathrm{j}}$ due to the magnetic field. If
$B_{x}=0$, calculate the magnetic field $\vec{B}$.

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Key Concepts

Lorentz Force Law

This law states that a charged particle moving in a magnetic field experiences a force given by F = q(v × B). It is pivotal in electromagnetism because it connects the particle's velocity, the magnetic field, and the resulting force, illustrating how charges interact with magnetic fields.

Vector Cross Product

The cross product of two vectors results in a third vector that is perpendicular to the plane formed by the original vectors. In the context of the Lorentz force, the cross product determines both the direction and the magnitude of the force acting on a moving charge in a magnetic field.

Charged Particles and Sign of Charge

Understanding the role of the charge sign is crucial since the direction of the force exerted by the magnetic field on a particle depends on whether the charge is positive or negative. For electrons, which have a negative charge, the force direction is opposite to the one predicted by the right?hand rule applied directly to the velocity and magnetic field vectors.

Uniform Magnetic Field

A uniform magnetic field is one that has the same magnitude and direction at every point in space. This uniformity simplifies calculations of the forces acting on charged particles and is a common assumption in introductory electromagnetism problems.

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