Question

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 $r_{2}$ $\left[r_{2} > r_{1}\right]$ 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

• A. $\frac{\mu_0}{2} n_1^2 i_1^2 \pi\left(r_2^2-r_1^2\right)$
• B. $\mu_0 n_1^2 i_1^2 \pi\left(r_2^2-r_1^2\right)$
• C. $\frac{\mu _{0}}{2}n_{1}^{2}i_{1}^{2}\pi \, r_{1}^{2}$
• D. $\frac{\mu _{0}}{2}n_{2}^{2}i_{2}^{2}\pi \, r_{2}^{2}$

Answer 1

Answer: (A)

The magnetic field is non-zero only in the region between the two solenoids, where $B=µ_{0} \, n_{2}i_{2}$
$\therefore $ Energy stored per unit volume
$=\frac{B^{2}}{2 \mu _{0}}=\frac{\mu _{0} n_{2}^{2} \, i_{2}^{2}}{2}$
The energy per unit length = energy per unit volume $\times $ area of cross section where $B \, \neq \, 0$
= $\frac{\mu _{0} n_{2}^{2} \, i_{2}^{2}}{2}\pi \, \left[r_{2}^{2} - \, r_{1}^{2}\right]$
$=\frac{\mu _{0} n_{1}^{2} \, i_{1}^{2}}{2}\left[\pi r_{2}^{2} - \, r_{1}^{2}\right]$ , since $n_{1}i_{1}=n_{2}i_{2}$