Concentric Spherical Capacitor

Consider two concentric metal spheres of finite thickness in a vacuum. The inner sphere has radii a₁ < a₂. The outer sphere has b₁ < b₂ (see Figure 1.1).

Figure 1.1

a) A charge Q₁ is put on the inner sphere and a charge Q₂ on the outer sphere. Find the charge density on each of the four surfaces. If Q₂ = -Q₁ what is the mutual capacitance of the system?

b) If the space between the spheres is filled with insulating material of dielectric constant ε₁, what are the surface charge densities and polarization surface charge densities for arbitrary Q₁ and Q₂ and the mutual capacitance for Q₂ = -Q₁.

SOLUTION

a) By using Gauss’s theorem and the fact that charges rearrange themselves so as to yield a zero electric field inside the conductors, we infer that all of Q₁ will reside on surface 2 of the inner sphere, -Q₁ on surface 3 of the outer sphere (no field in the interior of the outer sphere), and Q₁ + Q₂ on surface 4 of the outer sphere (see Figure 1.2). The surface charge

Figure 1.2

densities are straightforward to calculate as the charge divided by surface area:

If Q₂ = -Q₁, there is no charge on the external shell, simply Q₁ on surface 2 and –Q₁ on surface 3. The mutual capacitance may be calculated from ∆VC = Q₁, where ∆V is the difference in electric potential between the spherical shells. Again using Gauss’s theorem to calculate the magnitude E of the electric field between the shells, we have

b) The D field behaves like the E field before:

so, between the spheres

The real and polarization surface charge densities on surface 1 are still zero, and on surface 2, we find

Likewise, on the third and fourth surfaces,

Finally, the capacitance may be found: