# Capacitance of three concentric shells

• Kashmir
In summary: So the potential at radius a is just V=kq[1/a-1/c]. Similarly, the potential at radius c is just V=kq[1/c-1/a]. So the total potential at radius a+c is just V=kq[2/3-1/2]. This is just kq*[4*pi*epsilon], which is kq*[1]. So the capacitance is just C=Q/V.
Kashmir
Homework Statement
A friend of mine sent me this problem about finding the capacitance.
We have three concentric shells of radius a, b, c. And we've to find the capacitance between x and y.

I need help.

Thank you
Relevant Equations
C=Q/V

I want to calculate the capacitance of this system between the points x&y.
So suppose I give a charge Q to the outermost shell and -Q to the innermost shell. To find the capacitance C, I try to find the potential V between the outermost shell and innermost shell .
To find V ,I integrate the electric field and find it out to be V=kq[1/a -1/c] where k=1/(4 *pi*epsilon). Then I can find C =Q/V.
Is this approach correct.

Last edited:
This approach is correct. You should get the capacitance between the outer and inner shells the usual way as if the middle shell were not there. The middle shell is affecting nothing because it is an equipotential surface at the potential that would be there if there were vacuum between the shells.

A slightly more interesting problem might be to find the capacitance if a thick concentric shell were placed in between having wall thickness, say ##d=\frac{1}{2}(c-a)## with equal vacuum gaps between conductors.

Kashmir and berkeman
kuruman said:
This approach is correct. You should get the capacitance between the outer and inner shells the usual way as if the middle shell were not there. The middle shell is affecting nothing because it is an equipotential surface at the potential that would be there if there were vacuum between the shells.

A slightly more interesting problem might be to find the capacitance if a thick concentric shell were placed in between having wall thickness, say ##d=\frac{1}{2}(c-a)## with equal vacuum gaps between conductors.
Thank you. I’ll try your question :)

You don't need to do any integration. Just write down the potential at radius a due to the two shells. You know that the potential due to the outer shell there is the same as it is at radius c.

Kashmir

## What is the formula for the capacitance of three concentric spherical shells?

The capacitance of three concentric spherical shells can be calculated using the formula: $$C = \frac{4 \pi \epsilon_0}{\left(\frac{1}{R_1} - \frac{1}{R_2}\right) + \left(\frac{1}{R_2} - \frac{1}{R_3}\right)}$$, where $$R_1$$, $$R_2$$, and $$R_3$$ are the radii of the innermost, middle, and outermost shells, respectively, and $$\epsilon_0$$ is the permittivity of free space.

## How does the potential difference between the shells affect the capacitance?

The potential difference between the shells affects the capacitance in the sense that the capacitance is a measure of the ability to store charge for a given potential difference. The formula for capacitance inherently accounts for the potential difference through the radii of the shells. A larger potential difference for a given charge would result in a lower capacitance.

## Can the capacitance of three concentric shells be negative?

No, the capacitance of three concentric shells cannot be negative. Capacitance is a scalar quantity that represents the ability to store charge per unit potential difference, and it is always a positive value.

## What role does the permittivity of the medium play in the capacitance of three concentric shells?

The permittivity of the medium, denoted as $$\epsilon$$, directly affects the capacitance. In the formula $$C = \frac{4 \pi \epsilon}{\left(\frac{1}{R_1} - \frac{1}{R_2}\right) + \left(\frac{1}{R_2} - \frac{1}{R_3}\right)}$$, the permittivity $$\epsilon$$ (which can be $$\epsilon_0$$ for free space or $$\epsilon = \epsilon_0 \epsilon_r$$ for a dielectric material) appears in the numerator, meaning that a higher permittivity increases the capacitance.

## How does the spacing between the shells influence the overall capacitance?

The spacing between the shells influences the overall capacitance by affecting the terms in the denominator of the capacitance formula. Larger spacing (larger differences between the radii $$R_1$$, $$R_2$$, and $$R_3$$) results in a smaller denominator, thus increasing the capacitance. Conversely, smaller spacing reduces the capacitance.

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