# Equivalent capacitance for a system of spherical capacitors

The question demands equivalent capacitance of two concentric metallic shells with larger and smaller radii. b and a respectively. The inner sphere is just left open(it's not earthed or anything), and the outer sphere is connected to a battery of voltage V.

(a) represents the question and (b) is the solution diagram. The solution states that the inner sphere is not connected to anything so circuit is not complete(as shown in (b)). Hence the capacitance in the region between the two spherical shells is ignored. I couldn't understand this. According to me, there's supposed to be an electric field in region between the two shells. I'm unsure of the direction of field lines but I firmly believe there exists electric field in that region. And that would mean the presence electric field energy too . So HOW can we ignore the capacitance of the region between the two shells?! Although I can understand the solution diagram that circuit is not complete, but still it doesn't make any sense to me as to how that capacitance can be ignored. A simple yet convincing high school level explanation is appreciated.

Well, there's going to be a charge, $$Q$$, on $$B$$, so that:

$$V_B(b) \equiv -\int_{r=b}^{\infty}E(r)dr$$

$$V_B(b) =-\int_{r=b}^{\infty}\frac 1 {4\pi\epsilon_0}\frac Q {r^2}$$ $$V_b(B) = \frac 1 {4\pi\epsilon_0}\frac Q b = V$$

(where I choose $$V(\infty) = 0$$),

but even with that, the field inside the conductor is zero, whether there is a conducting sphere inside or not.

Even a uniformly charged dielectric sphere has no field inside (c.f., Gauss's Law)...there's no net charge.

Idk, seems like a strangely worded problem.

• That means no charge appears on the inner sphere? could you please convince me on this
– NPC
Commented Jun 28 at 18:28
• @NPC charge is conserved. How can it appear?
– JEB
Commented Jun 29 at 4:57
• I feel charge on the outer sphere can induce charges on inner sphere and electric field can be created in the region between the two shells... pls correct me
– NPC
Commented Jun 29 at 7:10
• @NPC clipped from a google for "field in a conductor": The intensity of the electric field in a charged conductor is zero; Because in a good conductor, the charges always go and settle on the conductor's surface, leaving zero charge inside and resulting in zero electric field.
– JEB
Commented Jun 29 at 11:55