# A cylindrical solenoid of length (ell) and radius (R) has (n) windings per unit length and carries

## Question:

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 \(I\) ?

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