Question

PQ is an infinite current carrying conductor. AB and CD are smooth conducting rods on which a conductor EF moves with constant velocity V as shown. The force needed to maintain constant speed of EF is (No gravity is there)

• A. $\frac{1}{vR}{\left[\frac{{\mu }_{0}IV}{2\pi }ln\left(\frac{b}{a}\right)\right]}^2$
• B. ${\left[\frac{{\mu }_{0}IV}{2\pi }ln\left(\frac{b}{a}\right)\right]}^2$
• C. ${\left[\frac{{\mu }_{0}IV}{2\pi }ln\left(\frac{b}{a}\right)\right]}^2\frac{V}{R}$
• D. ${\left[\frac{{\mu }_{0}IV}{3\pi }ln\left(\frac{2b}{a}\right)\right]}^2\frac{V}{R}$

Answer 1

The correct option is A $\frac{1}{vR}{\left[\frac{{\mu }_{0}IV}{2\pi }ln\left(\frac{b}{a}\right)\right]}^2$

Induced emf
${\int }_a^{b}Bvdx={\int }_a^b\frac{{\mu }_{o}l}{2\pi x}vdx{\left[\frac{{\mu }_{0}Iv}{2\pi }ln\left(\frac{b}{a}\right)\right]}^2\frac{v}{R}$
$\Rightarrow$ Induced emf
$=\frac{{\mu }_{0}lv}{2\pi }ln\left(\frac{b}{a}\right)$
$\Rightarrow$ Power dissipated
$=\frac{E^2}{R}$
Also, power = F.v $\Rightarrow F=\frac{E^2}{vR}$
$\Rightarrow F=\frac{1}{vR}{\left[\frac{{\mu }_{0}IV}{2\pi }In\left(\frac{b}{a}\right)\right]}^2$