Question

Two long coaxial solenoids each carry current I, but in opposite directions, as shown to the right. The solenoids are both of length L, which can be assumed to be very long, and are centered along the z-axis. The inner solenoid (radius a) has n turns per unit length, and the outer one (radius b) has the same n turns per unit length. 

\(\vec{B}\) inside the inner solenoid, \(=\mathbf{B}_{\text {inner }}=\mu_{0} I\left(n_{1}-n_{2}\right) \mathbf{z} \quad\) And \(\quad \mathbf{B}_{\text {middle }}=-\mu_{0} n_{2} I \hat{z}\)

a) Given

$$ W_{\operatorname{mag}}=\frac{1}{2 \mu_{0}} \int B^{2} d \tau $$

Find the energy stored in this region of space. Why is the answer different from having no inner solenoid?

b). What's the magnetic moment, \(\vec{m}\), of this system?

Answer 1