Question

Two electrons, r metres apart, move in parallel direction. What is the ratio of electric force to magnetic force experienced by an electron due to the other?

Answer 1

Two electrons move in parallel direction, at distance $r$ apart with speed $=v$ Let it move 'dx', So,

$dx=vdt$.

Current generated due to flow of charges:

$I=\frac{e}{dt}$, for both ....(1)

Electric force experienced, (from coulomb's law) by either charge-

$F_e=\frac{1}{L_1\pi {\in }_o}\frac{e^2}{r^2}$ ...(2)

The force per unit length due to generated magnetic field will be:-

$f=\frac{{\mu }_0{I}^2}{2\pi r}$

So, magnetic force on a electron,

$F_m=fdx=\frac{{\mu }_0{I}^2}{L_1\pi r^2}(dx{)}^2$ [assuming it provides centripetal force]

$\Rightarrow F_m=\frac{{\mu }_o{V}^2}{L_1\pi r^2}\left(e^2\right)$ ...(3) [From (1)]

Taking ratio of (2) & (3)

$\frac{F_e}{F_m}=\frac{1}{{\mu }_0{\in }_0{V}^2}=\frac{c^2}{V^2}$ $[as,c^2=\frac{1}{{\mu }_0{\in }_0}]$