Question

Write the expression for Lorentz magnetic force on a particle of charge ‘q’ moving with velocity ${\displaystyle \overrightarrow{v}}$ in a magnetic field${\displaystyle \overrightarrow{B}}$. Show that no work is done by this force on the charged particle.

Answer 1

 

Lorentz magnetic force,${\displaystyle \overrightarrow{F}=q(\overrightarrow{V}\times \overrightarrow{B})}$

Work done due to Lorentz force

${\displaystyle W=\overrightarrow{F}\cdot \overrightarrow{r}}$

 = ${\displaystyle q(\overrightarrow{V}\times \overrightarrow{B})\cdot \overrightarrow{r}}$

=${\displaystyle q[\overrightarrow{B}\cdot \overrightarrow{r}-\overrightarrow{V}\cdot \overrightarrow{r}]}$

${\displaystyle q[0-0]=0}$

as ${\displaystyle \overrightarrow{r}\perp \overrightarrow{B}\text{and}\overrightarrow{r}\perp \overrightarrow{v}}$

Hence, work done by the force on the charged particle will be zero.