You and a friend have a \(0.65-\mathrm{m}\) length of copper wire that has a diameter of \(4.115 \mathrm{~mm}\) and a wooden rod that is \(85 \mathrm{~mm}\) long and has a diameter of \(10 \mathrm{~mm}\). You're aiming to wind the wire around the rod about 40 times to form an inductor.

(a) What is the maximum inductance you can achieve with this device?

(b) Your friend solves this problem by reasoning that a cylindrically wound wire whose length greatly exceeds its radius is a solenoid, and therefore he can use the equation from Example 29.8 to determine the inductance:

\[ \begin{aligned} L & =\frac{\mu_{0} N^{2} A}{\ell} \\ & =\frac{\left(4 \pi \times 10^{-7} \mathrm{~T} \cdot \mathrm{m} / \mathrm{A}\right)\left(40^{2}\right) \pi(0.0020575 \mathrm{~m})^{2}}{0.65 \mathrm{~m}} \\ & =4.1 \times 10^{-8} \mathrm{H} . \end{aligned} \]

Evaluate your friend's work.