question_answer

Find the inductance of a unit length of two parallel wires, each of radius a, whose centres are at distance d apart and carry equal currents in opposite directions. Neglect the flux within the wire.

A) \[(p\to \tilde{\ }p\wedge )(\tilde{\ }p\to p)\]    

B) \[\frac{6!}{3!}\]

C) \[\underset{x\to -1}{\mathop{\lim }}\,\left( \frac{{{x}^{4}}+{{x}^{2}}=x+1}{{{x}^{2}}-x+1} \right)\frac{1-\cos (x+1)}{{{(x+1)}^{2}}}\]  

D) \[\sqrt{\frac{2}{3}}\]

Correct Answer: B

Solution :

Since, the wires are infinite, so the system of these two wires can be considered as closed rectangle of infinite length and breadth equal to d. Flux through strip \[-\frac{2}{{{x}^{2}}}+\frac{8}{{{(r-x)}^{2}}}=0\] The other wire produces the same result, so the total flux through the rectangle ABCD is \[\Rightarrow \] The total inductance of length l \[\frac{x}{r-x}=\sqrt{\frac{2}{8}}=\frac{1}{2}\Rightarrow x=\frac{r}{3}\] Inductance per unit length\[x=\frac{r}{3},\frac{{{d}^{2}}y}{d{{x}^{2}}}=\]