# Spherical Capacitor half filled with dielectric.

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In summary, the conversation discusses an isolated spherical capacitor with a liquid dielectric, and finding the capacitance, electric field, and surface charge densities for both the upper and lower halves. The capacitance can be calculated using a formula and the electric field can be found using the potential difference between the two halves. The surface charge densities can also be found using the dielectric constant and electric field. There is some uncertainty about combining the equations and finding the bound charge density.

#### Low_Profile

If anyone can help with this problem it would be greatly appreciated. I think I know what I'm doing, but am not sure of a couple things.
An isolated spherical capacitor has charge +Q on its inner conductor of radius r1 and charge -Q on its outer conductor of radius r2. half of the volume between the two conductors is then filled with a liquid dielectric of constant K.
a) find the capacitance of the capacitor.
b) find the magnitude of (electric field) E in the volume between the two conductors as a function of the distance r from the center of the capacitor. give answers for both the upper and lower halves of this volume.
c) find the surface density of free charge on the upper and lower halves of the inner and outer conductors.
d) what is the surface density of bound charge on the inner and outer surfaces of the dielectric.

So far I have attempted to calculate the capacitance using
C = 4*pi*epsilon*(r1*r2/r2 - r1)
for both the upper and lower halve of the sphere. This gives
c(upper) = 4*pi*epsilon_0*(r1*r2/r2 - r1)
c(lower) = 4*pi*k*(r1*r2/r2 - r1)
However I am not sure how to combine these.

for b) E_upper = Q/(4*pi*epsilon_0*r-sub-b^2)
E_lower = Q/(4*pi*k*epsilon_0*r-sub-b^2)

for c) D = epsilon_0*E
and this can be used to find surface density of free charge, but not sure how exactly.

for d) total charge density = Q/area
Can I take free charge density from total charge density to get bound charge density?

The potential difference V between the two conductors is the same for both the top and bottom halves, so think of the two halves as being two capacitors in parallel.

## What is a spherical capacitor half filled with dielectric?

A spherical capacitor half filled with dielectric refers to a type of capacitor that consists of two concentric spherical conductors, with a dielectric material (insulator) filling the space between them. The capacitor can be either completely filled with the dielectric or only half filled.

## What is the purpose of using a dielectric in a spherical capacitor?

The dielectric material in a spherical capacitor helps to increase the capacitance, or the ability to store electric charge, of the capacitor. This is because the dielectric reduces the electric field between the two conductors, allowing for more charge to be stored.

## How does a half-filled dielectric affect the capacitance of a spherical capacitor?

A half-filled dielectric in a spherical capacitor increases the capacitance compared to a completely filled capacitor. This is because the dielectric only partially reduces the electric field, allowing for more charge to be stored compared to a completely filled capacitor.

## What factors can affect the capacitance of a spherical capacitor half filled with dielectric?

The capacitance of a half-filled spherical capacitor can be affected by the size and distance between the two conductors, the type of dielectric material used, and the dielectric constant of the material. The capacitance also depends on the applied voltage and the frequency of the electric field.

## How is the capacitance of a spherical capacitor half filled with dielectric calculated?

The capacitance of a spherical capacitor half filled with dielectric can be calculated using the formula C = 4πεrε0a, where C is the capacitance, εr is the relative permittivity (dielectric constant) of the material, ε0 is the permittivity of free space, and a is the radius of the inner conductor.