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

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

Answer 1

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

A, Induced emf
${\int }_a^{b}Bvdx={\int }_a^b\frac{{\mu }_{0}I}{2\pi x}vdx$
$\Rightarrow$ Induced emf $=\frac{{\mu }_{0}Iv}{2\pi }ln\left(\frac{b}{a}\right)$
$\Rightarrow$ Power dissipated $=\frac{E^2}{R}$
Also, power = F.V $\Rightarrow F=\frac{E^2}{VR}$
$\Rightarrow F=\frac{1}{VR}{\left[\frac{{\mu }_{0}Iv}{2\pi }ln\left(\frac{b}{a}\right)\right]}^2$.