A conductor AB of length 10 cm at a distance of 10cm from an infinity long parallel conductor carrying a current 10 A…

Q: A conductor AB of length 10 cm at a distance of 10cm from an infinity long parallel conductor carrying a current 10 A. What work must be done to move AB to a distance of 20 cm if it carries 5 A ?

Sol: Force on a conductor at a distance x is $\large F = \frac{\mu_0 i_1 i_2 l}{2 \pi x}$

Work done to displace it through a small distance $\large dW = \vec{F}.\vec{dx} $

$\large dW = \frac{\mu_0 i_1 i_2 l}{2 \pi x} dx $

$\large W = \int_{0.1}^{0.2} \frac{\mu_0 i_1 i_2 l}{2 \pi x} dx $

$\large W = \frac{\mu_0 i_1 i_2 l}{2 \pi} [log_e x]_{0.1}^{0.2}$

$\large W = \frac{4\pi \times 10^{-7}\times 10 \times 5 \times 10 \times 10^{-2}}{2\pi}log_e 2 $

W = 0.693 × 10-6 J