Fundamentals of Plasma Physics

24 Chapter 1. Basic concepts

is the relationship between Tesla and Gauss, between particles per cubiccentimeter

and particles per cubic meter? What is magneticflux? If a circular loop of wire with

a break in it links a magneticflux of 29.83 Weber which increases at a constant rate to

aflux of 30.83 Weber in one second, what voltage appears across the break?

4. Solve Eq.(1.5) the ‘easy’ way by first proving using Gauss’ law to show thatthe solu-
   tion of
   ∇^2 φ=−

1

ε 0

δ(r)

is

φ=

1

4 πε 0 r

.

Show that this implies

∇^2

1

4 πr

=−δ(r) (1.39)

is a representation for the delta function. Then, use spherical polar coordinates and

symmetry to show that the Laplacian reduces to

∇^2 φ=

1

r^2

∂

∂r

(

r^2

∂φ

∂r

)

.

Explicitly calculate∇^2 (1/r)and then reconcile your result with Eq.(1.39). Using

these results guess that the solution to Eq.(1.5) has the form

φ=

g(r)

4 πε 0 r

.

Substitute this guess into Eq.(1.5) to obtain a differential equation forgwhich is trivial

to solve.

5. Solve Eq.(1.5) forφ(r)using a more general method which illustrates several im-
   portant mathematical techniques and formalisms. Begin by defining the 3D Fourier
   transform
   φ ̃(k) =

∫

drφ(r)e−ik·r (1.40)

in which case the inverse transform is

φ(r) =

1

(2π)^3

∫

dk ̃φ(k)eik·r (1.41)

and note that the Dirac delta function can be expressed as

δ(r) =

1

(2π)^3

∫

dkeik·r. (1.42)

Now multiply Eq.(1.5) by∫ exp(−ik·r)and then integrate over allr, i.e. operate with

dr. The term involving∇^2 is integrated by parts, which effectively replaces the∇

operator withik.

Show that the Fourier transform of the potential is

φ ̃(k) = qT

ǫ 0 (k^2 +λ−D^2 )

. (1.43)