Section 3.4 · Further exercises · 4 of 10
Exercise 3.4.4: Capacitor with a graded permittivity
About the further exercises of Section 3.4
The ten exercises below are in the spirit of Problem 3.4.1, the home exercises and the worked examples. They use only , Gauss's law , the boundary conditions, the bound charges, , and the image principle.
Capacitor with a graded permittivity
#cf. Problem 3.4.1
The gap of a parallel-plate capacitor of plate area is filled with a dielectric whose permittivity grows linearly,
The plate carries , the plate carries . Find (a) , and the capacitance, (b) the bound volume and surface charge densities, and check that the total bound charge is zero.
Adapted from: D. K. Cheng, Field and Wave Electromagnetics, 2nd ed., Problem P.3-30.
The medium is inhomogeneous, so we must not use or pull out of an integral. Instead: from the free charge alone (it is constant), then with inside the integral for the voltage. The bound charges follow from .
The plates are large, so . There is no free charge in the gap, so and is constant. The conductor condition at the plate gives . Then
The polarisation is
The bound volume charge density is
and on the faces of the dielectric ( at , at )
Total:
plus on the face : zero
lies between and , the values for the two extreme permittivities.
is the capacitance of infinitely many thin layers in series, . A nonuniform dielectric carries a bound volume charge even though there is no free charge in it (Figure 3.4.33).