Question

Loop $A$ of radius $(r<<R)$ moves toward loop $B$ with a constant velocity $V$ in such a way that their planes are always parallel. What is the distance between the two loops ($x$) when the induced emf in loop $A$ is maximum?

• A. $R$
• B. $\frac{R}{\sqrt{2}}$
• C. $\frac{R}{2}$
• D. $R(1-\frac{1}{\sqrt{2}})$
• E. answer required

Answer 1

The correct option is C $\frac{R}{2}$

${\varphi }_A=\frac{{\mu }_{o}i\pi R^2}{2\pi (R^2+x^2{)}^{3/2}}\pi r^2$

$E_A=-\frac{d\varphi }{dt}=\frac{{\mu }_{o}i\pi }{2}(-3/2)(R^2+x^2{)}^{-5/2}.2x(v)$

$E_A$ is maximum when $\frac{d{E}_A}{dx}=0$

or, $\frac{d}{dx}\frac{x}{(R^2+x^2{)}^{5/2}}=0$

or, $(R^2+x^2{)}^{5/2}=\frac{5x}{2}(R^2+x^2{)}^{3/2}2x=0$

Solving we get $x=\frac{R}{2}$