A proton of charge +e and mass m enters a uniform magnetic field B⃗=B î with an initial velocity

step 1 In this question, we have a proton, moving in a uniform magnetic view, and the magnetic field is

A proton of charge +e and mass m enters a uniform magnetic...

A proton of charge $+e$ and mass $m$ enters a uniform magnetic field $\vec{B}=B \hat{i}$ with an initial velocity $\vec{v}=v_{0 x} \hat{i}+v_{0 y} \hat{j} .$ Find an expression in unit-vector notation for its velocity $\vec{v}$ at any later time $t$.

Key Concepts

Lorentz Force

The Lorentz force is the fundamental force acting on a charged particle moving in a magnetic field, defined by F = q(v × B). This force is always perpendicular to both the velocity of the particle and the magnetic field, meaning that while the direction of the particle's velocity changes, its speed remains constant. This concept is central to understanding the dynamics of charged particles in electromagnetic fields.

Decomposition of Motion

In a uniform magnetic field, a charged particle's motion can be decomposed into two components: one that is parallel to the magnetic field and one that is perpendicular. The parallel component of the velocity experiences no magnetic force, thereby remaining unchanged, while the perpendicular component is subject to the Lorentz force and undergoes rotational or circular motion. This decomposition facilitates the analysis of the particle’s overall trajectory.

Cyclotron Frequency

Cyclotron frequency is the constant angular frequency with which the perpendicular component of a charged particle’s velocity rotates in a uniform magnetic field. Given by the expression ? = qB/m, this frequency determines the rate of circular motion and is instrumental in predicting the time evolution of the particle’s velocity vector. It is a key concept in the study of particle motion in magnetic fields, especially in applications like cyclotrons and mass spectrometers.

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