Question

A long wire carries a current $i$. A rod of length $l$ is moved with a velocity $v$ in a direction parallel to the wire as shown in figure (a). Find the motional emf induced in the rod :

• A. $\frac{{\mu }_{0}i}{2\pi x}lv$
• B. $\frac{{\mu }_{0}i}{2\pi }v{\log }_e\frac{x+l/2}{x-l/2}$
• C. $\frac{{\mu }_{0}i}{2\pi }v{\log }_e\frac{x-l/2}{x+l/2}$
• D. $\frac{{\mu }_{0}iv}{2\pi }{\log }_e\frac{l+x}{x}$

Answer 1

The correct option is B $\frac{{\mu }_{0}i}{2\pi }v{\log }_e\frac{x+l/2}{x-l/2}$

Consider small element dy at distance y from the long wire as shown in figure (b).

$B=\frac{{\mu }_{0}i}{2\pi y}$ and emf in element dy is $d\epsilon =Bv.dy$

Thus, $\epsilon ={\int }_{x-l/2}^{x+l/2}\frac{{\mu }_{0}iv}{2\pi }\frac{dy}{y}$$=\frac{{\mu }_{0}iv}{2\pi }{\log }_e\frac{x+l/2}{x-l/2}$