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Induced electric field in a solenoid

  1. Dec 13, 2012 #1
    1. The problem statement, all variables and given/known data

    A long solenoid with 1000 turns per meter and radius 2cm carries an oscillating current given by [itex](5A)\sin(100\pi t)[/itex]. What is the electric field induced at radius r=1cm from the axis of the solenoid? What is the direction of this electric field when the current is increasing counterclockwise in the coil?

    2. Relevant equations

    [tex]\oint\vec{E}\cdot d\vec{l}=-\frac{d\Phi_B}{dt}[/tex]
    [tex]\Phi_B=\oint\vec{B}\vec{dA}[/tex]and I have this on my formula sheet so I will start at this point (magnetic field inside a solenoid with n turns per unit length and current I[tex]B=\mu_0nI[/tex]

    3. The attempt at a solution

    So I think in this situation I can say [tex]\Phi_B=BA=(\mu_0nI)(\pi r^2)[/tex]So[tex]\frac{d\Phi_B}{dt}=\pi\mu_0nr^2\frac{dI}{dt}[/tex]
    [tex]\oint\vec{E}\cdot d\vec{l}=2E\pi r[/tex]
    So I got (ignoring the negative, just considering the magnitude)
    [tex]E=\frac{\pi\mu_0nr^2}{2\pi r}\frac{dI}{dt}= \frac{1}{2} \mu_0nr\frac{dI}{dt}=\frac{1}{2}\mu_0nr\omega I_{max}\cos(\omega t)[/tex]where [itex]\omega=100\pi[/itex] and [itex]I_{max}=5A[/itex]

    This makes sense to me but is not getting me the correct answer! Also I only reasoned that the electric field would be clockwise, but not really sure on that part. Does anyone see where I am going wrong?
  2. jcsd
  3. Dec 13, 2012 #2


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    Your work looks good to me. Did you convert r to SI units?
  4. Dec 13, 2012 #3
    ah sorry. this was the correct answer! the solution I was looking at messed up. what a waste of time typing that out! thank you for looking at this for me!
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