Magnetic field at \(A\) due to \(B\)

\(B=\frac{e v \sin 90^{\circ}}{d^2} \times \frac{\mu_0}{4 \pi}\)

\(B=\frac{e v}{d^2} \times \frac{\mu_0}{4 \pi}-(i)\)

Magnetic force on \(A\) is

\(F_B=e v B\)

From \((i)\), \(F_B=e v \frac{(e v)}{d^2} \times \frac{\mu_0}{4 \pi}\)

\(F_e=e t\)

\(=\frac{e Ke }{ d ^2}\)

\(\frac{F_B}{F_e}=\frac{e^2 v^2}{d^2} \times \frac{d^2}{e^2 k} \times \frac{\mu_0}{4 \pi}\)

\(=\frac{v^2}{\frac{1}{4 \pi \varepsilon_0} \times 4 \pi}\)

\(=\frac{v^2}{\frac{1}{\mu_0} \varepsilon_0}\)

\(\frac{F_B}{F_e}=\frac{v^2}{c^2}\)