Capacitance of a spherical capacitor

In summary, the conversation discusses calculating the capacity of a spherical capacitor with a dielectric material between the plates. The formula for capacitance is given and it is mentioned that the charge is constant while the potential varies due to distance. The final solution for capacitance is given and it is confirmed to be correct.
  • #1
Guillem_dlc
184
15
Homework Statement
A spherical capacitor is formed by two thin conductive layers, spherical and concentric, of radius [itex] R_1 [/itex] and [itex] R_2>R_1 [/itex], between which we have placed a dielectric material of relative permittivity [itex] \varepsilon_r [/itex]. Knowing that the inner layer has an [itex] Q [/itex] charge, idetermines the capacity of the capacitor and the total energy stored.
Relevant Equations
Gauss Law
When I try to do Gauss, the permeability is not always that of the free space, but it varies: up to a certain radius it is that of the void and then it is the relative one. How can I relate them? I'm trying to calculate the capacity of a spherical capacitor.

The scheme looks like this: inside I have the free space and between the plates of the capacitor I have the dielectric material.
67D98992-D759-4157-BF39-9E70AA815294.jpeg


The broken lines represent the Gaussian surface.
 
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  • #2
What is the result if it were space ( ε0) between the spherical shells?
 
  • #3
The vacuum doesn't matter because it contains no charge. The capacitor consists of two conducting plates with the space between them filled completely with the dielectric. Use a Gaussian surface completely inside the dielectric. Or you can find the capacitance with no dielectric between the shells and then multiply it by the dielectric constant.
 
  • #4
I think I have the solution. Is that right?
[tex]
\left.
\phi =\oint \vec{E}\cdot d\vec{S}=\oint E\cdot dS\cdot \underbrace{\cos 0}_1=E\oint dS=E\cdot S \atop
\phi =\dfrac{Q_{enc}}{\varepsilon_0 \varepsilon_r}=\dfrac{Q}{\varepsilon_0 \varepsilon_r}=\dfrac{\sigma \cdot S}{\varepsilon_0 \varepsilon_r}
\right\} E\cdot S=\dfrac{\sigma S}{\varepsilon_0 \varepsilon_r}\rightarrow E=\dfrac{\sigma}{\varepsilon_0 \varepsilon_r}=\dfrac{Q}{4\pi R^2 \varepsilon_0}
[/tex]
[tex] C=\dfrac{Q}{V_2-V_1} [/tex]
[tex] V_2-V_1=-\int_{R_1}^{R_2} \vec{E}\cdot d\vec{l}=-\int_{R_1}^{R_2}E\cdot \overbrace{dl\cdot \cos \theta}^{dR}=-\int_{R_1}^{R_2}\dfrac{Q}{4\pi R^2 \varepsilon_0 \varepsilon_r}dR=\dfrac{Q}{4\pi \varepsilon_0 \varepsilon_r}-\int_{R_1}^{R_2} \dfrac{1}{R^2}dR [/tex]
because [itex]Q[/itex] is constant as it has been transferred to us by an external field/generator. Therefore, it is invariant. [itex]V[/itex] varies due to distance. Then
[tex]V_2-V_1=\dfrac{Q}{4\pi \varepsilon_0 \varepsilon_r}\left( -\dfrac{1}{R_2}+\dfrac{1}{R_1}\right) \rightarrow C=\dfrac{4\pi \varepsilon_0}{\left( -\frac{1}{R_2}+\frac{1}{R_1}\right)}=\boxed{4\pi \varepsilon_0 \varepsilon_r\dfrac{R_2R_1}{R_2-R_1}} [/tex]
 
  • #5
That looks about right.
 
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Related to Capacitance of a spherical capacitor

1. What is the formula for calculating the capacitance of a spherical capacitor?

The formula for calculating the capacitance of a spherical capacitor is C = 4πε0r, where C is the capacitance, ε0 is the permittivity of free space, and r is the radius of the spherical capacitor.

2. How does the distance between the two plates affect the capacitance of a spherical capacitor?

The capacitance of a spherical capacitor is inversely proportional to the distance between the two plates. This means that as the distance between the plates decreases, the capacitance increases, and vice versa.

3. Can the capacitance of a spherical capacitor be negative?

No, the capacitance of a spherical capacitor cannot be negative. Capacitance is a measure of the ability of a capacitor to store charge, and it is always a positive value.

4. How does the material between the two plates affect the capacitance of a spherical capacitor?

The material between the two plates, also known as the dielectric material, affects the capacitance of a spherical capacitor by changing the permittivity value in the formula. Different materials have different permittivity values, which can increase or decrease the capacitance of the capacitor.

5. What is the unit of measurement for capacitance of a spherical capacitor?

The unit of measurement for capacitance of a spherical capacitor is Farads (F). This unit is named after Michael Faraday, a scientist who made significant contributions to the study of electricity and magnetism.

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