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

Answer 1

The correct option is A

$\frac{1}{VR}{\left[\frac{{\mu }_{0}IV}{2\pi }ln\left(\frac{b}{a}\right)\right]}^2$

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 $=F.V\Rightarrow F=\frac{E^2}{VR}$
$\Rightarrow F=\frac{1}{VR}{\left[\frac{{\mu }_{0}IV}{2\pi }ln\left(\frac{b}{a}\right)\right]}^2$