Question

Two coaxial long solenoids of equal lengths have current i₁, i₂, number of turns per unit length n₁, n₂ and radius r₁, r₂ respectively. If n₁i₁ = n₂i₂ and the two solenoids carry current in opposite sense, the magnetic energy stored per unit length is [r₂> r₁]-

• A. $\frac { \mu _ { 0 } } { 2 } n _ { 1 } ^ { 2 } i _ { 1 } ^ { 2 } \pi \left( r _ { 2 } ^ { 2 } - r _ { 1 } ^ { 2 } \right)$
• B.
• C. $\frac { \mu _ { 0 } } { 2 } \mathrm { n } _ { 1 } ^ { 2 } \mathrm { i } _ { 1 } ^ { 2 } \pi \mathrm { r } _ { 1 } ^ { 2 }$
• D.

Answer 1

Answer: (A)

$\frac { \mu _ { 0 } } { 2 } n _ { 1 } ^ { 2 } i _ { 1 } ^ { 2 } \pi \left( r _ { 2 } ^ { 2 } - r _ { 1 } ^ { 2 } \right)$

Answer 2

Magnetic field is non zero only in the region between the two solenoids, where B = µ₀ n₂i₂

\ energy stored per unit volume

=

The energy per unit length = energy per unit volume × area of cross section

where B ¹ 0 = $\frac { \mu _ { 0 } \mathrm { n } _ { 2 } ^ { 2 } \mathrm { i } _ { 2 } ^ { 2 } } { 2 }$ [p (r₂² – r₁²)]

= $\frac { \mu _ { 0 } \mathrm { n } _ { 1 } ^ { 2 } \mathrm { i } _ { 1 } ^ { 2 } } { 2 }$ [p (r₂² – r₁²), since n₁i₁ = n₂i₂