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**1. Homework Statement**

A wire of length [itex]\ell[/itex], radius [itex]r_1[/itex], and resistivity [itex]\rho[/itex] is tightly wound in a single layer into the shape of a solenoid with circular cross section of radius [itex]r_2[/itex]. Assume it is an ideal solenoid. A DC voltage [itex]V[/itex] 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 [itex]V_{rms}[/itex] and frequency [itex]f[/itex], what is the rms current [itex]I_{rms}[/itex] through the solenoid?

**2. Homework Equations**

[tex]R = \frac{\rho \ell}{A}[/tex]

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.

[tex]

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

[/tex]

[tex]

N = \frac{\ell}{2 \pi r_2}

[/tex]

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

[tex]

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

[/tex]

b.

[tex]

\mathrm{Magnetic flux linkage} = N\Phi = LI

[/tex]

[tex]

L = \frac{ABN}{I}

[/tex]

Plugging in, I got

[tex]

L = \frac{\mu_0 \ell}{2 \pi}

[/tex].

c.

[tex]

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

[/tex]

[tex]

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

[/tex]

[tex]

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

[/tex]

[tex]

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

[/tex]

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.