W9_parallel_resonance.eps

86 Week 2: Continuous Charge and Gauss’s Law

where it is presumed that everybody knows how to integrate to evalute the area of a cylindrical
surface of radiusrand lengthLandknows the result^37. Note that I indicate explicitly that theflux
through the end caps is zero even though the field there may not be.

The total chargeQS 1 inside this cylinder iszeroby inspection – the fingers and toes thing. That
was easy! Now we write Gauss’s law:

φe=

∮

S 1

E~·ˆrdA=Er(2πrL) =QS^1

ǫ 0

= 0 (113)

and solve forEr:

Er(2πrL) = 0

=

0

2 πrL

Er = 0 forr < a (114)

We’ve just shown thatin generalthe electric field of a cylindrical shell of chargevanishesinside.

Outside the shell we draw asecondcylindrical Gaussian surfaceS 2 with lengthLatr > a.
Again, the field must be constant and normal to all points on this surface from symmetry, again the
flux through the end caps must be zero even though the field on theend caps may not be. The flux
integral isidentical:

φe =

∮

S 2

E~·rˆdA

= φcaps+Er

∫

C

dA

= Er(2πr)L (115)

and in fact it willalwaysbe this algebraic form for a cylindrical problem, to the point where wewill
get bored writing this line out umpty times doing homework. Don’t let that stop you! Do it every
time, as when you know something well enough to be slightly bored writing it out, that’s just about
perfect, isn’t it?

Again we can count up the charge insideS 2 on the thumbs of one hand. It is the total charge on
the shellinside the Gaussian surface of lengthL!Which is, in fact (noting thatdAfor a cylindrical
shell of radiusaisadθ dz):

QS 2 =

∫

S

σ 0 dA=

∫ 2 π

0

dθ

∫L/ 2

−L/ 2

aσ 0 dz

= 2πaLσ 0 (116)

which wecouldhave done using our heads instead of calculus, but again this way youget to see how
to do a two dimensional integral that separates into two trivial onedimensional integrals.

Finally, we write out Gauss’s law and solve forEr:

φe=Er(2πrL) =

QS 2

ǫ 0

Er =

2 πaLσ 0

2 πLǫ 0

1

r

= σ^0

ǫ 0

a

r

=^2 kλ^0

r

(117)

(^37) Think of the label of a soup can. Use mental scissors to snip, snip, snip it off. Unroll it in your mind. It is 2πr
long andLwide.