8.10 And in Conclusion..

Back

I have just got you to stick your toes in the water as far as PDEs are concerned.

There are people who make a lifetime’s work of solving Laplace’s

8.10. AND IN CONCLUSION.. 197

Equation for progressively more complicated boundary conditions, and they

feel their time is well spent. There are applications of PDEs in mining, working

out from gravity surveys where the body (the ore body) is buried, in Electromagnetism,

in Quantum Physics, in studying waves in every medium you

can imagine and a good few you can’t, in the twisting and bending of solids,

and the list goes on. I finish with a little exercise which should convince you

that the material covered in the course has a certain value:

Exercise 8.10.1. And God Said ...

div E = 0 div H = 0

curl E = −

1

c

@H

@t

curl H =

1

c

@E

@t

The above are Maxwell’s Equations relating the electric field E and the magnetic

field H. c is a constant which depends on the electrical and magnetic

properties of free space and which can be measured by fixed physical apparatus.

Show that For any field F on R3, curl curl F = grad div F - r2F.

Now using Maxwell’s Equations show that:

1.

r × (r × E) = −

1

c2

@2E

@t2

2.

r × (r × H) = −

1

c2

@2H

@t2

3.

r2E =

1

c2

@2E

@t2

4.

r2H =

1

c2

@2H

@t2

5. Deduce that there are electromagnetic waves that travel at a speed of

c.

...And there was light!