Question

Two protons move parallel to each other with an equal velocity $v=300km$ $s^{-1}$. Find the ratio of forces of magnetic and electric interaction of the protons. Give your answer after multiplying by ${10}^6$.

Answer 1

The force of magnetic induction is given by
$\overrightarrow{F_{mag}}=e(\overrightarrow{v}\times \overrightarrow{B});\overrightarrow{B}=\frac{{\mu }_0}{2\pi }\frac{e(\overrightarrow{v}\times \overrightarrow{r})}{r^3}$
$\overrightarrow{{\therefore F}_{mag}}=\frac{{\mu }_0}{2\pi }\times \frac{e^2}{r^3}[\overrightarrow{v}\times (\overrightarrow{v}\times \overrightarrow{r}\left)\right]$
$=\frac{{\mu }_0}{2\pi }\times \frac{e^2}{r^3}\left[\right(\overrightarrow{v}.\overrightarrow{r})\times \overrightarrow{v}-(\overrightarrow{v}.\overrightarrow{v})\times \overrightarrow{r}]$
$=\frac{{\mu }_0}{4\pi }\frac{e^2}{r^3}\times (-v^2\overrightarrow{r})$
$\therefore \frac{\left|F_{mag}\right|}{\left|F_{ele.}\right|}=v^2{\mu }_0{\epsilon }_0={\left(\frac{v}{c}\right)}^2=1.00\times {10}^{-6}$