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 to $EF$ is

• A. $\frac{1}{vR}{\left[\frac{{\mu }_{0}Iv}{2\pi }\ln \frac{\left(b\right)}{\left(a\right)}\right]}^2$
• B. $\frac{v}{R}{\left[\frac{{\mu }_{0}Iv}{2\pi }\ln \frac{\left(a\right)}{\left(b\right)}\right]}^2$
• C. $\frac{v}{R}{\left[\frac{{\mu }_{0}Iv}{2\pi }\ln \frac{\left(b\right)}{\left(a\right)}\right]}^2$
• D. None of these

Answer 1

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

$B_x=\frac{{\mu }_0}{2\pi }\frac{I}{x}$

$de=B_{x}vdx=\frac{{\mu }_0}{2\pi }\frac{I}{x}vdx$

$e={\int }_a^{b}de=\frac{{\mu }_{0}Iv}{2\pi }\ln \left(\frac{b}{a}\right)$

$i=\frac{e}{R}=\frac{{\mu }_{0}Iv}{2\pi R}\ln \left(\frac{b}{a}\right)$

$=$ Induced current

$dF=\left(i\right)\left(dx\right)B_x$

$=\left[\frac{{\mu }_{0}Iv}{2\pi R}\ln \left(\frac{b}{a}\right)\right]\left[\frac{{\mu }_0}{2\pi }\frac{I}{x}\right]dx$

$F={\int }_a^{b}dF=\frac{1}{vR}{\left[\frac{{\mu }_{0}Iv}{2\pi }\ln \frac{\left(b\right)}{\left(a\right)}\right]}^2$