Two coaxial long solenoids of equal lengths have currents `i_(1) and i_(2)`, the number of turns per unit length `n_(1) and n_(2)` and radii `r_(1) and r_(2)[r_(2)gt r_(1)]` respectively. If `n_(1)i_(1)=n_(2)i_(2)` and the two solenoids carry currents in the opposite sense, the magnetic energy stored per unit length is

Options

A. `(mu_(0))/(2)n_(1)^(2)pi(r_(2)^(2)-r_(1)^(2))`
B. `mu_(0)n_(1)^(2)i_(1)^(2)pi(r_(2)^(2)r_(1)^(2))`
C. `(mu_(0))/(2)n_(1)^(2)i_(1)^(2)pi r_(1)^(2)`
D. `(mu_(0))/(2)n_(2)^(2)i_(2)^(2)pi r_(2)^(2)`

Explanation

Magnetic field is non zero only in the region between the two solenoids, where `B=mu_(0)n_(2)i_(2)`
`:.` energy stored per unit volume `=(B_(2))/(2mu_(0))=(mu_(0)n_(2)^(2)i_(2)^(2))/(2)`
The energy per unit length.=energy per unit volume x area of cross section where `Bne0`
`=(mu_(0)n_(2)^(2)i_(2)^(2))/(2)[pi(r_(2)^(2)-r_(1)^(2))]=(mu_(0)n_(1)^(2)i_(1)^(2))/(2)[pi(r_(2)^(2)-r_(1)^(2))]` , since `n_(i)i_(1)-n_(2)i_(2)`