# A solenoid

1. Homework Statement

A wire of length $\ell$, radius $r_1$, and resistivity $\rho$ is tightly wound in a single layer into the shape of a solenoid with circular cross section of radius $r_2$. Assume it is an ideal solenoid. A DC voltage $V$ is placed across the ends of the solenoid.

a. What is the magnitude of the magnetic field inside the solenoid?
b. What is the self-inductance of the solenoid?
c. If the DC voltage source is replaced with an AC source with rms voltage $V_{rms}$ and frequency $f$, what is the rms current $I_{rms}$ through the solenoid?

2. Homework Equations

$$R = \frac{\rho \ell}{A}$$
Ampere's law
Ohm's law

3. The Attempt at a Solution
a.

I found current and turn density in terms of the given variables.

$$I = \frac{V}{R} = \frac{\pi V r_1^2}{\rho \ell}$$
$$N = \frac{\ell}{2 \pi r_2}$$

I plugged everything into the equation for resistance, and got the following expression for magnetic field.

$$B = \frac{\mu_0 V r_1^2}{2 \rho \ell r_2}$$

b.

$$\mathrm{Magnetic flux linkage} = N\Phi = LI$$
$$L = \frac{ABN}{I}$$
Plugging in, I got
$$L = \frac{\mu_0 \ell}{2 \pi}$$.

c.

$$V(t) = L\frac{dI}{dt}$$

$$V_{rms}\sqrt{2} \sin{2\pi f t} dt = L dI$$

$$-\frac{V_{rms}\cos{2\pi f t}}{\sqrt{2} \pi f} = L I(t)$$

$$I_{rms} = \frac{2V_{rms}}{\mu_0 f \ell}$$

I have absolutely no idea if I did any of this problem correctly, so I would appreciate it if someone could check what I did.