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

Options

A. `(mu_(0))/(2pi) In(( d- a)/(a))`
B. `(mu_(0))/(pi) In(( d- a)/(a))`
C. `(3mu_(0))/(pi) In(( d- a)/(a))`
D. `(mu_(0))/(3pi) In(( d- a)/(a))`

Answer

B

Explanation

Since the wires are infinite, so the system of these two wires can be considered as a closed rectangle of infinite length and breadth equal to d. flux through the strip:
`phi = int_(a)^(d-a) (mu_(0)I)/(2pir) (ldr) = (mu_(0)Il)/(2pi) In ((d - a)/(a))`

The other wire produces the same result, so the total flux through the dotted rectangle is
`phi_(total) = (mu_(0)lI)/(pi) In((d - a)/(a))`
The total inducatance of length `l`
`L = (phi_(total))/(I) = (mu_(0)l)/(pi) In ((d - a)/(a))`
Inductance per unit length `= (L)/(l) = (mu_(0))/(pi) In ((d - a)/(a))`