Magnetic Force and electric forces Between two Moving Charges

Suppose two charges q₁ and q₂ are moving with velocities v₁ and v₂ respectively.Let r be the distance between the particles at any instant.

Two forces would be acting on each charges at this particular instant

a) Electric Force which is given by

$F_e=\frac{1}{4\pi {\epsilon }_0}\frac{q_1{q}_2}{r^2}$

b) Magnetic Forces due to the instantanous magnetic field setup by the moving charges

Which can be calculated as below

The instantanous magnetic field setup by q₁ at the point of space where q₂ is situated is given by

$B=\frac{{\mu }_0}{4\pi }\frac{q_1{v}_1\sin \theta }{r^2}$

where θ is the angle between v₁ and r

The magnetic force on the charge q₂ due to this magnetic field

F=q₂v₂Bsinϕ

where ϕ is the angle between v₂ and B

So magnetic force will be

$F_m=\frac{{\mu }_0}{4\pi }\frac{q_1{q}_2{v}_1{v}_2\sin \theta \sin \phi }{r^2}$

if v₁=v₂=v and θ=ϕ=90⁰, then

$F_m=\frac{{\mu }_0}{4\pi }\frac{q_1{q}_2{v}^2}{r^2}$

So

$\frac{F_m}{F_e}={\mu }_0{\epsilon }_0{v}^2$

We know that

${\mu }_0{\epsilon }_0=1/c^2$

Where c is the velocity of the light in free space

So

$\frac{F_m}{F_e}={\left(\frac{v}{c}\right)}^2$

In general v <<<<c So magnetic force are weaker then electric forces.