Chapter 41: Problem 32

Question: Figure 20 (Chapter 41) shows a parallel plate capacitor being charged. (a) Show that the Poynting vector ${\bf S}$ points everywhere radially inward into the cylindrical volume. (b) Show that the rate at which energy flows into this volume, calculated by integrating the Poynting vector over the cylindrical boundary of this volume, is equal to the rate at which stored electrostatic energy increases; that is

$$\int {\bf S}\cdot d{\bf A}~=~ Ad {d\over dt}\left( {1\over 2}\epsilon_0 E^2\right) \ \ \ ,$$ (10)

where Ad is the volume of the capacitor and ${1\over 2}\epsilon_0 E^2$is the energy density for all points within that volume.

Solution: The cylindrical symmetry of the problem tells us that the magnetic field is invariant under axial rotations. As the electric flux penetrating any flat surface is maximized when the surface is perpendicular to the flux, we see that the magnetic field is as shown in Fig.2, chapter 40.

Applying Amperes law in the gap (where I=0), gives

$$\int {\bf B}\cdot d{\bf l}~=~ \mu_0\epsilon_0 {d\over dt} \Phi_E \ \ ,$$ (11)

which gives

$$B(r)~=~{1\over 2}\mu_0\epsilon_0 r {d\over dt} E \ \ \ ,$$ (12)

for the uniform electric field in the volume of the capacitor. The magnitude of the Poynting vector is then

$$\vert{\bf S}\vert~=-~{1\over\mu_0}E B~=~{\epsilon_0 r\over 4} {d\over dt}\left(E^2\right) \ \ \ .$$ (13)

Integrating this over the surface of the volume of the capacitor gives

$$\int {\bf S}\cdot d{\bf A}~=~{\epsilon_0 R\over 4} {d\over dt}\le... ...ight) 2 \pi R d~=~A d {d\over dt}\left( {1\over 2}\epsilon_0 E^2\right) \ \ \ ,$$ (14)

where $A~=~\pi R^2$.