A cylindrical solenoid of length \(\ell\) and radius \(R\) has \(n\) windings per unit length and carries a current \(I\).

(a) Use the inductance expression \(L=\left(\mu_{0} N^{2} A\right) / \ell\) from Example 29.8 and Eq. 29.25, \(U^{B}=L I^{2} / 2\), to derive an expression for the magnetic potential energy in the solenoid as a function of the current and the solenoid's dimensions.

(b) Derive this same expression by beginning with Eq. 29.30, \(U^{B}=\int u_{B} d V\). 

Data from Example 29.8

What is the inductance of a solenoid of length \(\ell\) that has \(N\) windings, each of cross-sectional area \(A\), when the current through the device is \(I\) ?

Transcribed Image Text:

UB = LI. (29.25)