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A solenoid

  • Thread starter Saketh
  • Start date
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2
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.
 

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