A proton moves perpendicular to a uniform magnetic field 𝐁...

A proton moves perpendicular to a uniform magnetic field $\mathbf{B}$ at a speed of $1.00 \times 10^{7} \mathrm{m} / \mathrm{s}$ and experiences an acceleration of $2.00 \times 10^{13} \mathrm{m} / \mathrm{s}^{2}$ in the positive $x$ direction when its velocity is in the positive $z$ direction. Determine the magnitude and direction of the field.

Key Concepts

Lorentz Force

The Lorentz force describes the force acting on a charged particle moving in a magnetic field. It is given by the equation F = q(v × B), where q is the charge, v is the velocity, and B is the magnetic field. This force is always perpendicular to both the particle's velocity and the magnetic field, leading to motion that is often circular or helical in nature.

Centripetal Acceleration

When a charged particle moves in a uniform magnetic field with its velocity perpendicular to the field, the magnetic force acts as a centripetal force. This results in centripetal acceleration, which is directed toward the center of the circular path. It is calculated using a = v²/r, where v is the speed of the particle and r is the radius of the circle.

Uniform Magnetic Field

A uniform magnetic field is characterized by having the same magnitude and direction at every point within a given region. This uniformity greatly simplifies the analysis of a charged particle’s motion, as the forces involved become constant in direction and magnitude relative to the particle's motion, typically resulting in predictable circular or helical trajectories.

Right-Hand Rule

The right-hand rule is an essential tool for determining the direction of the force exerted on a positive charge moving in a magnetic field. By directing the fingers of the right hand in the direction of the particle's velocity and curling them towards the direction of the magnetic field, the thumb then points in the direction of the force acting on the particle.

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