Near the equator, the Earth's magnetic field points almost...

Near the equator, the Earth's magnetic field points
almost horizontally to the north and has magnitude
$B=0.50 \times 10^{-4} \mathrm{T} .$ What should be the magnitude and
direction for the velocity of an electron if its weight is to be
exactly balanced by the magnetic force?

Key Concepts

Lorentz Force

The Lorentz force is the force exerted on a charged particle moving in a magnetic field. Its magnitude is given by F = qvB sin(?), where q is the charge, v is the velocity, B is the magnetic field strength, and ? is the angle between the velocity and magnetic field vectors. This concept is central to understanding how magnetic fields interact with moving charges.

Gravitational Force

Gravitational force is the force that a mass experiences due to gravity, defined by F = mg, where m is the mass and g is the acceleration due to gravity. In problems involving equilibrium, the gravitational force is one of the key forces that must be balanced by another force, such as the magnetic force.

Force Equilibrium

Force equilibrium occurs when all the forces acting on an object balance each other, resulting in no net force and no acceleration. In scenarios where a magnetic force is used to counteract gravitational force, the condition for equilibrium is that the magnitude of the magnetic force equals the gravitational force.

Vector Cross Product and Direction

The magnetic force on a moving charge depends on the cross product of the velocity and magnetic field vectors, which means its magnitude involves the sine of the angle between them and its direction is perpendicular to both. Understanding this vector relationship is crucial for determining the correct orientation of motion needed to achieve a specific force balance.

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