Maxwell’s Equations

1. Maxwell’s Equations Chapter 32, Sections 9, 10, 11 Maxwell’s Equations Electromagnetic Waves Chapter 34, Sections 1,2,3

2. The Equations of Electromagnetism (at this point …) Gauss’ Law for Electrostatics Gauss’ Law for Magnetism Faraday’s Law of Induction Ampere’s Law

3. The Equations of Electromagnetism ..monopole.. Gauss’s Laws 1 ? 2 ...there’s no magnetic monopole....!!

4. The Equations of Electromagnetism .. if you change a magnetic field you induce an electric field......... Faraday’s Law 3 Ampere’s Law 4 .......is the reverse true..?

5. Look at charge flowing into a capacitor E B Ampere’s Law Here I is the current piercing the flat surface spanning the loop.

6. Look at charge flowing into a capacitor E B Ampere’s Law Here I is the current piercing the flat surface spanning the loop. E For an infinite wire you can deform the surface and I still pierces it. But something goes wrong here if the loop encloses one plate of the capacitor; in this case the piercing current is zero. B Side view:(Surface is now like a bag:)

7. Look at charge flowing into a capacitor E It must still be the case that B around the little loop satisfies B E where I is the current in the wire. But that current does not pierce the surface. B What does pierce the surface? Electric flux - and that flux is increasing in time.

8. Look at charge flowing into a capacitor E B E B Thus the steady current in the wire produces a steadily increasing electric flux. For the sac-like surface we can write Ampere’s law equivalently as

9. Look at charge flowing into a capacitor E B The best way to write this result is E B Then whether the capping surface is the flat (pierced by I) or the sac (pierced by electric flux) you get the same answer for B around the circular loop.

10. Maxwell-Ampere Law E B This result is Maxwell’s modification of Ampere’s law: Can rewrite this by defining the displacement current (not really a current) as Then

11. Maxwell-Ampere Law E B This turns out to be more than a careful way to take care of a strange choice of capping surface. It predicts a new result: A changing electric field induces a magnetic field This is easy to see: just apply the new version of Ampere’s law to a loop between the capacitor plates with a flat capping surface: B x x x x x x x x x x x x

12. Maxwell’s Equations of Electromagnetism Gauss’s Law for Electrostatics Gauss’s Law for Magnetism Faraday’s Law of Induction Ampere’s Law

13. Maxwell’s Equations of Electromagnetism Gauss’s Law for Electrostatics Gauss’s Law for Magnetism Faraday’s Law of Induction Ampere’s Law These are as symmetric as can be between electric and magnetic fields – given that there are no magnetic charges.

14. Maxwell’s Equations in a Vacuum Consider these equations in a vacuum: no charges or currents

15. Maxwell’s Equations in a Vacuum Consider these equations in a vacuum: no charges or currents These integral equations have a remarkable property: a wave solution

16. E(x, t) = EP sin (kx-t) B(x, t) = BP sin (kx-t) ˆ j ˆ z Plane Electromagnetic Waves Ey Bz This pair of equations is solved simultaneously by: c x as long as

17.  F(x) x  F(x) v x Static wave F(x) = FP sin (kx + ) k = 2   k = wavenumber  = wavelength Moving wave F(x, t) = FP sin (kx - t)  = 2  f  = angular frequency f = frequency v =  / k

18. F v Moving wave F(x, t) = FP sin (kx - t) x At time zero this is F(x,0)=Fpsin(kx).

19. F v Moving wave F(x, t) = FP sin (kx - t) x At time zero this is F(x,0)=Fpsin(kx). Now consider a “snapshot” of F(x,t) at a later fixed time t.

20. F v Moving wave F(x, t) = FP sin (kx - t) x At time zero this is F(x,0)=Fpsin(kx). Now consider a “snapshot” of F(x,t) at a later fixed time t. Then F(x, t) = FP sin{k[x-(/k)t]} This is the same as the time-zero function, slide to the right a distance (/k)t.

21. F v Moving wave F(x, t) = FP sin (kx - t) x At time zero this is F(x,0)=Fpsin(kx). Now consider a “snapshot” of F(x,t) at a later fixed time t. Then F(x, t) = FP sin{k[x-(/k)t]} This is the same as the time-zero function, slide to the right a distance (/k)t. The distance it slides to the right changes linearly with time – that is, it moves with a speed v= /k. The wave moves to the right with speed /k

22. E(x, t) = EP sin (kx-t) B(x, t) = BP sin (kx-t) ˆ j ˆ z Plane Electromagnetic Waves These are both waves, and both have wave speed /k.

23. E(x, t) = EP sin (kx-t) B(x, t) = BP sin (kx-t) ˆ j ˆ z Plane Electromagnetic Waves These are both waves, and both have wave speed /k. But these expressions for E and B solve Maxwell’s equations only if Hence the speed of electromagnetic waves is

24. E(x, t) = EP sin (kx-t) B(x, t) = BP sin (kx-t) ˆ j ˆ z Plane Electromagnetic Waves These are both waves, and both have wave speed /k. But these expressions for E and B solve Maxwell’s equations only if Hence the speed of electromagnetic waves is Maxwell plugged in the values of the constants and found

25. E(x, t) = EP sin (kx-t) B(x, t) = BP sin (kx-t) ˆ j ˆ z Plane Electromagnetic Waves These are both waves, and both have wave speed /k. But these expressions for E and B solve Maxwell’s equations only if Hence the speed of electromagnetic waves is Maxwell plugged in the values of the constants and found

26. E(x, t) = EP sin (kx-t) B(x, t) = BP sin (kx-t) ˆ j ˆ z Plane Electromagnetic Waves These are both waves, and both have wave speed /k. But these expressions for E and B solve Maxwell’s equations only if Hence the speed of electromagnetic waves is Maxwell plugged in the values of the constants and found Thus Maxwell discovered that light is electromagnetic radiation.

27. E(x, t) = EP sin (kx-t) B(x, t) = BP sin (kx-t) ˆ j ˆ z Ey Plane Electromagnetic Waves Bz c • Waves are in phase. • Fields are oriented at 900 to one another and to the direction of propagation (i.e., are transverse). • Wave speed is c • At all times E=cB. x

28. The Electromagnetic Spectrum infra -red ultra -violet Radio waves g-rays m-wave x-rays