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Cylindrical Capacitor: Gauss Law

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A capacitor consists of two concentric long cylindrical conductors of radii a and c, where c > a, and each is of negligible thickness. The space between the conductors is filled with two layers of dielectric materials. Using cylindrical co-ordinates with z-axis along the axis of the cylindrical conductors, the space a<r<b is filled with an LIH (linear, isotropic and homogeneous) dielectric of relative permittivity e1 and the space b<r<c with a second LIH dielectric of relative permittivity e2 . The cylinders are sufficiently long that end effects can be neglected. The charge per unit length of the inner conducting cylinder (r=a) is lembda, and that on the outer conducting cylinder (r=c) is -lembda.

a) Use the integral version of Gauss's law to find the electric field E and the electric displacement D in the region a < r < c. Hence calculate the electrostatic field energy stored in a length l of the capacitor.

b) Hence find the capacitance of a length l of the capacitor.

(Please see the attachment also)

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a) For the region a < r < b (region 1) : Let us consider a cylindrical Gaussian surface of length L shown in dotted red in the fig. and apply Gauss theorem : ∫E.ds = Q/ε where Q is the total charge enclosed by the Gaussian surface. Here, Q = charge on length L of the inner cylindrical plate = λL

Let E1 be the radially outwards electric field on the curved face of the ...

Solution Summary

The expert examines cylindrical capacitors for Gauss Laws.

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Dielectrics and Infinite Electric Cylinders

1. The space between the spaces of a parallel plate capacitor is filled with two slabs of linear dielectric material. The slabs have different dielectric constants but the same length L, width W and thickness d. (note the area of the top (or bottom) of the capacitor is 2*L*W). Slab 1 has a dielectric constant of e1=2 and slab 2 has a dielectric constant of e2=1.5. The voltage on the top plate is Vo and the bottom plate is grounded.

a) What is the electric Field E in each slab?
b) What is the Dielectric displacement D in each slab?
c) What is the polarization P in each slab?
d) What is the magnitude of Sf, the free surface charge adjacent to each slab?
e) What is the magnitude of Sb, the bound surface charge density in each slab?
f) What is the magnitude of Vb, bound volume charge density in each slab?
g) What is the capacitance of the system and how does it compare to the capacitance of the system with no dielectrics?

See the attachment for an illustration.

2) We have an infinite electric cylinder of radius R that has a polarization

Pbar=kr^2 rhat (a vector)

a) What is the bound volume density in the body of the cylinder
b) What is the bound surface density at r=R
c) Use Gauss' Law to find the electric field inside and outside the cylinder
d) Use a releationship between

E, D, and P to find the electric fields without using Gauss' Law.

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