Consider a spherical capacitor whose inner plate is a metal sphere of radius a and a spherical metal shell of inner radius c = 5.5 a. The space within the shell is filled with two different dielectric materials, as shown in the figure (but not to scale). Specifically, the space between radius a and radius b= 2.2 a, is filled with a material with dielectric constant x₁ = 1.7, and the space between radius b and radius c, is filled with a material with dielectric constant x₂= 3.7. This spherical capacitor is charged so the surface of inner plate carries a uniform charge +Q, and the inner surface of the outer plate has a uniform charge -Q. Note: All of the quantities mentioned above are in units of coulombs and meters, as appropriate. NT (a) What is the capacitance C of this system. Express your answer in terms of the capacitance Co of an air-filled capacitor of the same plate geometry? Co (b) What is the electric potential energy stored in this capacitor? Express your answer in terms of the electric potential energy Up stored in an air-filled capacitor of the same plate geometry with charges +Q and -Q on its plates? U₂ (c) What is the electric potential difference between the inner metal sphere and the interface (at radius b) between the two dielectrics? Express your answer in terms of AVC, the total potential difference between the inner and outer plates of the dielectric-filled capacitor.
Consider a spherical capacitor whose inner plate is a metal sphere of radius a and a spherical metal shell of inner radius c = 5.5 a. The space within the shell is filled with two different dielectric materials, as shown in the figure (but not to scale). Specifically, the space between radius a and radius b= 2.2 a, is filled with a material with dielectric constant x₁ = 1.7, and the space between radius b and radius c, is filled with a material with dielectric constant x₂= 3.7. This spherical capacitor is charged so the surface of inner plate carries a uniform charge +Q, and the inner surface of the outer plate has a uniform charge -Q. Note: All of the quantities mentioned above are in units of coulombs and meters, as appropriate. NT (a) What is the capacitance C of this system. Express your answer in terms of the capacitance Co of an air-filled capacitor of the same plate geometry? Co (b) What is the electric potential energy stored in this capacitor? Express your answer in terms of the electric potential energy Up stored in an air-filled capacitor of the same plate geometry with charges +Q and -Q on its plates? U₂ (c) What is the electric potential difference between the inner metal sphere and the interface (at radius b) between the two dielectrics? Express your answer in terms of AVC, the total potential difference between the inner and outer plates of the dielectric-filled capacitor.
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Transcribed Image Text:Consider a spherical capacitor whose inner plate is a metal sphere of radius a and a spherical metal shell of inner radius c = 5.5 a. The space within the shell is filled with two different dielectric materials,
as shown in the figure (but not to scale). Specifically, the space between radius a and radius b = 2.2 a, is filled with a material with dielectric constant x₁ = 1.7, and the space between radius b and radius
c, is filled with a material with dielectric constant K₂= 3.7. This spherical capacitor is charged so the surface of inner plate carries a uniform charge +Q, and the inner surface of the outer plate has a
uniform charge -Q.
Note: All of the quantities mentioned above are in units of coulombs and meters, as appropriate.
K2
K1
(a) What is the capacitance C of this system. Express your answer in terms of the capacitance Co of an air-filled capacitor of the same plate geometry?
Co
(b) What is the electric potential energy stored in this capacitor? Express your answer in terms of the electric potential energy Up stored in an air-filled capacitor of the same plate geometry with charges
+Q and -Q on its plates?
Uo
(c) What is the electric potential difference between the inner metal sphere and the interface (at radius b) between the two dielectrics? Express your answer in terms of AVC, the total potential difference
between the inner and outer plates of the dielectric-filled capacitor.
AVC
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