Gauss' law for D

Gauss' law, spherical symmetry

Problem:

A spherical capacitor with conducting surfaces of radii R1 and R2 has a material of dielectric constant
ε(r) = ε0(R1/r)2
between the spheres.
(a)  Find the capacitance C of the capacitor.
(b)  If the charge on the capacitor is Q find the total energy stored in the capacitor.

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Solution:

Problem:

A spherical conductor of radius a is surrounded with a dielectric shell of outer radius b.  The dielectric constant varies with radius as K = 1 + n(r - a), where n is a constant.  A charge Q is placed on the conductor.
(a)  Find the electric displacement and the electric field at all points in space.
(b)  Find the distribution of bound charges in the interior of and on the surface of the dielectric shell.
(c)  Find the total energy stored in the system.

Solution:

Problem:

A linear dielectric sphere of radius a and dielectric constant ke carries a uniform charge density ρ, surrounded by vacuum.
(a)  Find E and D inside and outside the sphere.
(b)  Find the energy U of the system.

Solution:


Gauss' law, cylindrical symmetry

Problem:

A capacitor is made of two concentric cylinders of radii r1 and r2 (r1 < r2) and length L >> r2.
The region between r1 and r3 = (r1r2)½ is filled with a circular cylinder of length L and dielectric constant k.
The remaining volume is an air gap.
(a)  What is the capacitance?
(b)  What are the values of E, P, and D at a radius r inside the dielectric (r1 < r < r3)?  Assume a potential difference V  between r1 and r2.
(c)   How much mechanical work must be done to remove the dielectric cylinder while maintaining this constant potential difference between r1 and r2?
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Solution:

Problem:

A 500 m length of high-voltage cable is undergoing electrical testing.  The cable consists of two coaxial conductors, the inner of 5 mm diameter and the outer of 9 mm internal diameter.  The space between the conductors is filled with polythene which has a relative permittivity of 2 and which can withstand electric field strength of 60 MVm-1.
(a)  Find the maximum voltage which can be applied between the conductors and the energy stored in the cable at this voltage.
(b)  If the cable is to be discharged to a safe level of 50 V in 1 minute, what value of resistance must be connected across it?  What is the maximum power and the total energy dissipated in the resistance?

Solution:


Gauss' law, planar symmetry

Problem:

The dielectric of a parallel plate capacitor has a permittivity that varies as ε1 + ax, where x is the distance from one plate.  The area of each plate is A and their spacing is s.
Assume a surface charge density ±σfree for the plates.
(a)  Find the capacitance.
(b)  Assume ε1 + ax varies from ε1 to 2ε1.  Find P from D and E for that case.
(c)  Find the polarization charge density ρp.

Solution:

Problem:

The plates of an isolated parallel-plate capacitor have area A and are separated by a distance d << A½.  Let one plate be located in the y-z plane at x = 0 and the other plate at x = d.  The plate at x = 0 carries a charge +Q and the plate at x = d carries a charge -Q.  Dielectric 1 with permittivity ε = ε0 + ax fills the region between the plates from x = 0 to x = d/2, and dielectric 2 with permittivity ε = ε0 + a(d - x) fills the region between the plates from x = d/2 to x = d.  Let a = ε0/d.

(a)  Find the electric displacement and the electric field between the plates.
(b)  Find the voltage across the plates.
(c)  Find the distribution of bound charges in the interior of and on the surfaces of both dielectrics.

Solution: