Question

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

Answer 1

Component of velocity parallel to $\overrightarrow{B}$ i.e., $v_x$ will remain unchanged $v_y\hat{j}$ and $B_j$ will rotate the particle in y-z plane ($\perp$ to $\overrightarrow{B}$).
At the beginning direction to magnetic force is $-\hat{k}$ [from the relation, $\overrightarrow{F}=q(v_y\hat{j}\times B\hat{i})$]
In the time t, particle rotates and angle $\theta =\omega t=(\frac{Bq}{m})t$ from its original path.
In the figure we can set that, y-component of velocity at time t is $v_{y}cos\theta$ and z-component is $-v_{y}sin\theta$.