Simple Nature - Light and Matter

The total field is

Ez=

∫

dEz

= 2πσk z

∫b

0

rdr

(

r^2 +z^2

)3/2

= 2πσk z

− 1

√

r^2 +z^2

∣∣

∣∣

r=b

r=0

= 2πσk

(

1 −

z

√

b^2 +z^2

)

The result of example 15 has some interesting properties. First,

we note that it was derived on the unspoken assumption ofz >0.

By symmetry, the field on the other side of the disk must be equally

strong, but in the opposite direction, as shown in figures e and g.

Thus there is a discontinuity in the field atz= 0. In reality, the

disk will have some finite thickness, and the switching over of the

field will be rapid, but not discontinuous.

At large values ofz, i.e.,zb, the field rapidly approaches the

1 /r^2 variation that we expect when we are so far from the disk that

the disk’s size and shape cannot matter (homework problem 2).

g/Example 15: variation of the field (σ>0).

A practical application is the case of a capacitor, f, having two

parallel circular plates very close together. In normal operation, the

charges on the plates are opposite, so one plate has fields pointing

into it and the other one has fields pointing out. In a real capacitor,

600 Chapter 10 Fields