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A solenoid

  1. Jan 23, 2007 #1
    1. The problem statement, all variables and given/known data

    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. Relevant 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.
     
  2. jcsd
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