Question

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

Answer 1

Force experienced in magnetic field

$\overrightarrow{F}=q(\overrightarrow{V}\times \overrightarrow{B})$

$⟹\overrightarrow{F}=q[-\overrightarrow{V_i}.\overrightarrow{B}]\hat{k}$

where $\overrightarrow{k}$ is unit vector in $z-$direction.

Acceleration in $z-$direction:

$\overrightarrow{a}=\frac{\overrightarrow{f_z}}{m_e}$ where $m_e=$mass of charge

$=\overrightarrow{a}=\left(q/m_e\right)\times [-\overrightarrow{V_i}.\overrightarrow{B}]$

Newtons laws of motion states $\overrightarrow{v}=\overrightarrow{u}+\overrightarrow{a}t$

$⟹\overrightarrow{v_z}=\overrightarrow{a}t$

$⟹\overrightarrow{v}=v_x\hat{i}+v_y\hat{j}+v_z\hat{k}$

$=v_x\hat{i}+v_y\hat{j}-\left(\frac{q}{m}\right).(-\overrightarrow{V_i}.\overrightarrow{B})$