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Displacement current inside spherical capacitor

  1. Jan 29, 2013 #1
    You have a spherical capacitor with inner conductor radius a and outer conductor with radius b. The capacitor is filled with a perfect homogeneous dielectric of permittivity ε and is connected to a low-frequency time-harmonic voltage v(t)=V[itex]_{0}[/itex]cos(ωt). Find the displacement current density vector at an arbitrary point in the dielectric.

    Equations:

    C=[itex]\frac{εS}{r}[/itex] ; S-area of the plates; r-distance

    i(t)=C[itex]\frac{dv}{dt}[/itex]


    Approach:

    for starters I subbed the capacitance equation into the current equation and achieved this result:

    i(t)=[itex]\frac{εS}{r}[/itex]*[itex]\frac{dv}{dt}[/itex]

    taking the first derivative of the voltage and subbing it into the equation gives me:

    i(t)=[itex]\frac{εS}{r}[/itex]*-V[itex]_{0}[/itex]ωsin(ωt)

    Now I divide both sides of the equation by S in order to get the current density. I then integrate this equation with respect to r from the inner radius to the outer radius:

    J[itex]_{d}[/itex](t)=-[itex]\frac{εV_{0}sin(ωt)}{ln(b/a)}[/itex]

    More or less I don't know if I computed this correctly so any help would be appreciated.
     
  2. jcsd
  3. Jan 29, 2013 #2

    rude man

    User Avatar
    Homework Helper
    Gold Member

    1. Compute C for your capacitor. Don't forget ε.
    2. Use i = CdV/dt to get the total current.
    3. Find the current density at any point r, a < r < b, by dividing the total current by the area of the surface at r.

    For (1), use Gauss's law and C = q/V. V = ∫abE(r)dr.
     
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