Spherical capacitor with 2 dielectrics.

In summary, the conversation discusses finding the capacitance of a spherical capacitor with two different homogeneous dielectrics arranged concentrically. The solution involves applying the spherical capacitance formula and adding the expressions for the inner and outer spheres. One person also attempts to solve the problem using the fact that E is the field for a sphere of charge, and calculates the voltage using a circulation integral. After simplifying their answer, they arrive at the same solution as the previous method.
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Homework Statement


The problem is on page 40 of this PDF:
http://web.mit.edu/8.02t/www/802TEAL3D/visualizations/coursenotes/modules/guide05.pdf
Find the capacitance of a spherical capacitor with 2 different homogeneous dielectrics arranged concentrically.

The Attempt at a Solution



In that document they simply apply the spherical capacitance formula and get the expressions for two capacitors, the inner and outer sphere, then they add their inverses as its pretty much 2 capacitors in series.

I've been trying to solve the problem "my way" by starting straight from the fact that E is the field for a sphere of charge, and calculate the voltage: circulation integral from the surface of the inner terminal to the outer shell/terminal.

So I have to add the following integrals

[itex] V = \int^a_b Q/4\pi\epsilon_1r^2\,dr + \int^b_c Q/4\pi\epsilon_2r^2\,dr[/itex]

Is this correct? Because if I do the math/LCM and divide Q by it etc, my expression for the capacitance differs. Am I assuming something invalid with my integral? I know E should be constant between the cap terminals, but the field lines circulate through 2 different dielectrics so I thought splitting up the integral was reasonable.

EDIT: OOPS nevermind, lack of simplifying my answer made me think it was wrong when I had arrived to the same answer as that pdf.

Just want to make sure in case though, did I calculate V correctly?
 
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FAQ: Spherical capacitor with 2 dielectrics.

1. What is a spherical capacitor with 2 dielectrics?

A spherical capacitor with 2 dielectrics is a type of capacitor that consists of two concentric spherical conductors with a gap between them, filled with two different dielectric materials. The dielectrics act as insulators and help to increase the capacitance of the capacitor.

2. How does a spherical capacitor with 2 dielectrics work?

A spherical capacitor with 2 dielectrics works by storing electric charge on its conductive surfaces. The two dielectrics between the conductors have different permittivity values, which leads to a larger electric field and an increased capacitance. When a voltage is applied, the electric charge is stored on the conductors and the dielectrics help to maintain a stable electric field.

3. What is the capacitance of a spherical capacitor with 2 dielectrics?

The capacitance of a spherical capacitor with 2 dielectrics can be calculated using the formula C = 4πεrε0 / (1/r1 - 1/r2), where εr is the relative permittivity of the dielectric material, ε0 is the permittivity of free space, and r1 and r2 are the radii of the inner and outer conductors, respectively.

4. How does the dielectric material affect the capacitance of a spherical capacitor with 2 dielectrics?

The dielectric material has a significant impact on the capacitance of a spherical capacitor with 2 dielectrics. A material with a higher permittivity will increase the capacitance, while a material with a lower permittivity will decrease the capacitance. Additionally, the thickness of the dielectric layer also affects the capacitance, with a thicker layer resulting in a higher capacitance.

5. What are some common applications of a spherical capacitor with 2 dielectrics?

Spherical capacitors with 2 dielectrics have various applications in the field of electronics and electrical engineering. They are commonly used in high-voltage power supplies, radio frequency circuits, and in the construction of capacitors for electronic devices. They are also used in particle accelerators for storing and manipulating high-energy charged particles.

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