Displacement current inside spherical capacitor

[Mr.Tibbs]

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_{0}cos(ωt). Find the displacement current density vector at an arbitrary point in the dielectric.

Equations:

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

i(t)=C\frac{dv}{dt}


Approach:

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

i(t)=\frac{εS}{r}*\frac{dv}{dt}

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

i(t)=\frac{εS}{r}*-V_{0}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_{d}(t)=-\frac{εV_{0}sin(ωt)}{ln(b/a)}

More or less I don't know if I computed this correctly so any help would be appreciated.

 

[rude man]

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_{0}cos(ωt). Find the displacement current density vector at an arbitrary point in the dielectric.

Equations:

C=\frac{εS}{r} ; S-area of the plates; r-distance
Irrelevant. You do not have a parallel-plate capacitor.


i(t)=C\frac{dv}{dt}
Approach:

.

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 = ∫_a^bE(r)dr.