Capacitance – Capacitance of Two Coaxial Cylinders with/without Grounding the Outer Cylinder

capacitance

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.

Best Answer

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.

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