VI. Electromagnetic Waves

1. VI. Electromagnetic Waves All the important physics in electromagnetism can be expressed in Maxwell’s Equations with interesting consequences.

2. VI–1 Maxwell’s Equations

3. Main Topics • Generalized Ampères Law. • Maxwell’s Equations. • Production of Electromagnetic Waves. • Electromagnetic Waves Qualitatively.

4. Generalized Ampère’s Law I • We have learned that lineintegral of magneticinduction over any closed path depends on the total current which is enclosed by this path. • But is it valid generally? • If we are charging a capacitor, experiment shows that magnetic field can be measured near to the capacitor as if a current went through it. But charges don’t pass through the capacitor.

5. Generalized Ampère’s Law II • If a theory is in a contradictionwith the experiment it must be improved or changed! • We have to accept that whateverhappens in the capacitor when it is being charged behaveslike a current because it producesmagneticfield. It must be a newtype of current, since it is definitely not movement of charges.

6. Generalized Ampère’s Law III • It is, of course, the electricfield what changes in the charged (or discharged) capacitor. So we define a new type of current which we call a displacement (or shift) current which we attribute to the change-in-time of the electric field, more accurately the electric flux. To see this we have to understand what enclosing means.

7. Generalized Ampère’s Law IV • Up to now, when we used Ampere’s law we usually used a circular closed path and have taken into account the total current which passed the circular surface bounded by it. • Generally, we can consider surface of any shape, bounded by the enclosed path. The currents, which would enter the surface but would not pass through our path would have to leave the surface somewhere so they would not be counted in.

8. Generalized Ampère’s Law V • The fact that some of these surfaces can also pass between the plates of the capacitor means that what passes there through them must be equivalent to the electric current. And now since we are interested in an electric field passing through surfaces, we are, in fact interested in electric flux, which deals also with the mutual directions.

9. Generalized Ampère’s Law VI • The existence of displacement current actually means an important symmetry between electric and magnetic fields since not only changes of magnetic fields produceelectric fields but also the changes of electric fields produce magnetic fields! • Thanks to this symmetry electromagnetic waves and also WE do exist!

10. Generalized Ampère’s Law VII • The displacement current for a parallel plate capacitor can be calculated simply from the definition of its capacity and the current: Q = CV = (0A/d)(Ed) = 0AE I = dQ/dt = d(0AE)/dt = 0 de/dt • This conclusion is valid generally so the Ampere’s law has a new form.

11. Generalized Ampère’s Law VII • For instance if we are charging a capacitor from a power source V0 , through a resistor R. The current decreases exponentially from an initial value I0 = V0/R so the initial change of the electric field dE/dt = I0/0A and the magnetic field B outside: B = 00r02/2r dE/dt = 00r02/2r I0/0r02  B = 0I0/2r

12. Maxwell’s Equations • Now we are ready to write the Maxwell’s equations. • There are several types of them and several levels of generality. So exactly we shall deal with Maxwell’s equations in integral form valid for vacuum.

13. Maxwell’s Equation I • The first ME is the Gauss law of electrostatics. It means that: • Sources of electric field – chargesexist. • The electric field lines begin in positive charges (or infinity) andend in negative charges (or infinity). • The field of a single charge decreases as 1/r2.

14. Maxwell’s Equation II • The second ME is the Gauss law of magnetism it means that: • Separate sources of magnetic field – magnetic monopolesdo not exist. • The magnetic field lines are closed. • The field of an elementary current decreases as 1/r2.

15. Maxwell’s Equation III • The third ME is the Faraday’s law of electro magnetic induction. It means that: • The electric field is produced also by changes-in-time of magnetic field. • In the absence of varying magnetic field the magnetic field is conservative and a scalarpotential exists.

16. Maxwell’s Equation IV • The fourth ME is the generalized Ampères law. It means that: • The magnetic field is produced by electric currents or by changes-in-time of electric field.

17. Maxwell’s Equations V • ME plus the Lorentz force are the basic equations for all electromagnetism. • They have many important consequences. • E.g. we see that we can treat electricity and magnetism separately only if they do not vary in time. • The most important outcome is the existence of electromagnetic waves.

18. Planar Electromagnetic Waves • Important solution of ME is a planar wave. If it moves in positive x direction with a speed c, the field can be described as: E = Ey =E0sin(kx - t) B = Ez =B0sin(kx - t) • E and B are in phase • c, E, B form right turning (hand) system • wave number: k = 2/ • angle frequency:  = 2f • speed of the wave: c = f = /k

19. Production of Electromagnetic Waves • Since changes of electric field produces magnetic field and vice versa these fields once generated can continue to exist and spread into the space. • This can be illustrated using a simple dipole antenna and an AC generator. • Planar waves will exist only far from the antenna where the dipole field disappears.

20. Homework • Chapter 31 – 1, 2, 3, 4, 7, 12, 13, 24, 25, 40

21. Things to read and learn • Chapter 32 – 1, 2, 3, 4 • Try to understand all the details of the scalar and vector product of two vectors! • Try to understand the physical background and ideas. Physics is not just inserting numbers into formulas!

22. Generalized Ampere’s Law • Ienclsum of all enclosed currents taking into account their directions and • 0de/dtis the displacement current due to change-in-time of the electric flux. ^

23. Maxwell’s Equations I • . ^