A particle of mass 10 g and charge 80 μC moves through a uniform magnetic field, in a region wh

step 1 In this question, we have a particle that is moving to the right at a constant speed of 20 kilom

A particle of mass 10 g and charge 80 μC moves through a...

A particle of mass $10 \mathrm{~g}$ and charge $80 \mu \mathrm{C}$ moves through a uniform magnetic field, in a region where the free-fall acceleration is $-9.8 \hat{\mathrm{j}} \mathrm{m} / \mathrm{s}^{2} .$ The velocity of the particle is a constant $20 \mathrm{i} \mathrm{km} / \mathrm{s}$. which is perpendicular to the magnetic field. What, then, is the magnetic field?

Key Concepts

Vector Cross Product

The vector cross product is used to determine the direction and magnitude of the magnetic force on a moving charge. Since the magnetic force is determined by the cross product of the velocity and the magnetic field vectors, understanding this mathematical operation is essential for analyzing problems where the direction of forces plays a critical role.

Force Equilibrium

For a charged particle moving with constant velocity, the net force acting on it must be zero. When additional forces such as gravity are present, another force—in this case, the magnetic force—must exactly counterbalance the gravitational force for the particle’s speed and direction to remain unchanged.

Uniform Magnetic Field

A uniform magnetic field has the same magnitude and direction at every point in the region of interest. This uniformity simplifies calculations and analysis, as the force on a charged particle will be consistent throughout the particle's trajectory, making it easier to predict and measure its motion.

Lorentz Force

The Lorentz force describes the force exerted on a charged particle moving in electric and magnetic fields, and it is given by F = q(v × B). This force is always perpendicular to both the particle’s velocity and the magnetic field, which is a fundamental concept in electromagnetism and is crucial for understanding how charged particles behave in magnetic fields.

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