Simple Nature - Light and Matter

of theinwardfield contributed by the outside sphere is

|E−|=

{

0, r<b

k q/r^2 , r>b

.

In the region outside the whole capacitor, the two fields are equal

in magnitude, but opposite in direction, so they cancel. We then

have for the total field

|E|=







0, r<a

k q/r^2 , a<r<b

0, r>b

,

so to calculate the energy, we only need to worry about the region

a<r<b. The energy density in this region is

dUe

dv

=

1

8 πk

E^2

=

k q^2

8 π

r−^4.

This expression only depends onr, so the energy density is con-

stant across any sphere of radiusr. We can slice the region

a< r < binto concentric spherical layers, like an onion, and

the energy within one such layer, extending fromrtor+ dris

dUe=

dUe

dv

dv

=

dUe

dv

(area of shell)(thickness of shell)

= (

k q^2

8 π

r−^4 )(4πr^2 )(dr)

=

k q^2

2

r−^2 dr.

Integrating over all the layers to find the total energy, we have

Ue=

∫

dUe

=

∫b

a

k q^2

2

r−^2 dr

=−

k q^2

2

r−^1

∣

∣∣

∣

b

a

=

k q^2

2

(

1

a

−

1

b

)

608 Chapter 10 Fields