Capacitance of two coaxial cylinders when the outer cylinder is/isn't grounded

I know how to solve the following question, but I have trouble understanding the difference between two situations:

• A cylindrical wire of charge of radius a and charge per unit length λ is at the center of a thin cylindrical conducting shell of radius b. What is the capacitance per unit length of this configuration?

I know that I should consider a Gaussian surface and calculate V to calculate C. But I'm confused about the following:

1. If the outer cylinder is not grounded, then its net charge should be zero. Therefore, the electric field still exists outside the cylinder. Is it true? It doesn't seem like a real capacitor to me.

2. If the outer cylinder is grounded, or a charge per unit length −λ is distributed on it, then the electric field outside the cylinder should be zero. Right?

I think that the second situation is the only one where we have an actual capacitor. The situation where no net charge is placed on the outer cylinder confuses me.

• If you charge a capacitor and completely disconnect it from the charging circuit where does the charge reside? Sep 30, 2023 at 10:03
• @Farcher Well, on its plates! Sep 30, 2023 at 10:15
• And where on the plates, inside or outside? Sep 30, 2023 at 10:42
• @Farcher I think we usually consider the plates to be very thin. So it doesn't really make sense to me to say "inside" the plates, cuz we'll have surface charge density. Anyways, even if we don't consider them to be thin, I think the correct one is "outside". Sep 30, 2023 at 10:56
• Please refer to another post here. Sep 30, 2023 at 16:05

2 Answers

Let us call the inner cylinder C1 and outer one C2

As soon as we make such a set up, and give λ (charge per unit length) to the inner cylinder C1, it will induce equal and opposite charge on the inner plater of outer cylinder C2. Now we can derive various cases

1. If outer cylinder is not grounded, the inner plate induced charge on C2 will contribute to formation of capacitor while the opposite of induced charge will stay on the outside plate of C2.
2. If outer cylinder is grounded, the inner plate induced charge on C2 will contribute to formation of capacitor while the opposite of induced charge will get grounded. This can also happen if we connect wires and form a circuit.

In each of the above case, an electric field lies in the region between the cylindrical surfaces of the capacitor C1 and C2.

There might electric field even outside C2 (as in case 1.), but that only adds to the complexity of the question and is not our concern since we define capacior as an energy storing device and for this capacitor, energy is stored between outer plate of C1 and inner plate of C2 in the form of electric field.

To sum up, your main concern was to understand induced charge.

• Thank you so much for your detailed response, Ayesha! So basically the thing that matters here is that the induced charges on the inner side of C2 contribute to the formation of a capacitor. So, we somehow should NOT consider C2 as infinitely thin... Oct 1, 2023 at 10:35
• I'm glad my response helped! You have summarised this right. And off course there is no such thing as infinitely thin, even if it is one atom thin, its charges can polarize. But I would like to add one thing and that is – it can be induced charge on inner side of C2 or it can also be just the distribution of charge on inner C2 (need not always be induced) Oct 2, 2023 at 7:14

The capacitance of this arrangement of cylinders has nothing to do with any charges on it. So it does not depend where you have potential zero. To calculate the capacitance you put a potential difference V on the two sides and calculate the charge Q , or place a charge Q and calculate the potential difference V. But what you use as V has nothing to do with the capacitance.

• Thank you for your reply. I understand what you mean, but the electric charge on the outer cylinder makes me confused a little bit. Are there any charges on it? What is the difference between when the outer cylinder is grounded and when it is not? Sep 30, 2023 at 12:18