1. Electromagnetic Waves
Chapter 24
Electromagnetic Waves

2. Unpolarized visible light X-ray
Electromagnetic waves cover a broad spectrum of wavelengths, with waves in various wavelength ranges having distinct properties. These photos of the Crab Nebula show different structure for observations made with waves of various wavelengths. The photos (clockwise starting from the upper left) were taken with x-rays, unpolarized visible light, radio waves, and visible light passing through a polarizing filter.
Radio waves
Polarized visible light

3. 24.1 Electromagnetic Waves, Introduction
Electromagnetic (EM) waves permeate our environment
EM waves can propagate through a vacuum
Much of the behavior of mechanical wave models is similar for EM waves
Maxwell’s equations form the basis of all electromagnetic phenomena

4. Conduction Current
A conduction current is carried by charged particles in a wire
The magnetic field associated with this current can be calculated by using Ampère’s Law:
The line integral is over any closed path through which the conduction current passes

5. Conduction Current, cont.
Ampère’s Law in this form is valid only if the conduction current is continuous in space
In the example, the conduction current passes through only S1 but not S2
This leads to a contradiction in Ampère’s Law which needs to be resolved

6. James Clerk Maxwell
1831 – 1879
Developed the electromagnetic theory of light
Developed the kinetic theory of gases
Explained the nature of color vision
Explained the nature of Saturn’s rings

7. Displacement Current
Maxwell proposed the resolution to the previous problem by introducing an additional term called the displacement current
The displacement current is defined as

8. Displacement current The electric flux through S2 is EA
S2 is the gray circle
A is the area of the capacitor plates
E is the electric field between the plates
If q is the charge on the plates, then Id = dq/dt
This is equal to the conduction current through S1

9. Displacement Current
The changing electric field may be considered as equivalent to a current
For example, between the plates of a capacitor
This current can be considered as the continuation of the conduction current in a wire
This term is added to the current term in Ampère’s Law

10. Ampère-Maxwell Law
The general form of Ampère’s Law is also called the Ampère-Maxwell Law and states:
Magnetic fields are produced by both conduction currents and changing electric fields

11. 24.2 Maxwell’s Equations, Introduction
In 1865, James Clerk Maxwell provided a mathematical theory that showed a close relationship between all electric and magnetic phenomena
Maxwell’s equations also predicted the existence of electromagnetic waves that propagate through space
Einstein showed these equations are in agreement with the special theory of relativity

12. Maxwell’s Equations Gauss’ Law (electric flux)
Gauss’ Law for magnetism
Faraday’s Law of induction
Ampère-Maxwell Law
The equations are for free space
No dielectric or magnetic material is present

13. Lorentz Force
Once the electric and magnetic fields are known at some point in space, the force of those fields on a particle of charge q can be calculated:
The force is called the Lorentz force

14. 24.3 Electromagnetic Waves
In empty space, q = 0 and I = 0
Maxwell predicted the existence of electromagnetic waves
The electromagnetic waves consist of oscillating electric and magnetic fields
The changing fields induce each other which maintains the propagation of the wave
A changing electric field induces a magnetic field
A changing magnetic field induces an electric field

15. Plane EM Waves
We assume that the vectors for the electric and magnetic fields in an EM wave have a specific space-time behavior that is consistent with Maxwell’s equations
Assume an EM wave that travels in the x direction with the electric field in the y direction and the magnetic field in the z direction

16. Plane EM Waves, cont The x-direction is the direction of propagation
Waves in which the electric and magnetic fields are restricted to being parallel to a pair of perpendicular axes are said to be linearly polarized waves
We assume that at any point in space, the magnitudes E and B of the fields depend upon x and t only

17. Figure 24.4: As a plane wave moving in the +x direction passes through a rectangular path of width dx lying in the xy plane, the electric field in the y direction varies from E→ to E→ + dE→. This construction allows us to evaluate the line integral E→ of over the perimeter of the rectangle.

18. Equations of the Linear EM Wave
From Maxwell’s equations applied to empty space, E and B are satisfied by the following equations
These are in the form of a general wave equation, with
Substituting the values for mo and eo gives c = x 108 m/s

19. Solutions of the EM wave equations
The simplest solution to the partial differential equations is a sinusoidal wave:
E = Emax cos (kx – wt)
B = Bmax cos (kx – wt)
The angular wave number is k = 2 p / l
l is the wavelength
The angular frequency is w = 2 p ƒ
ƒ is the wave frequency

20. Ratio of E to B The speed of the electromagnetic wave is
Taking partial derivations also gives

21. Properties of EM Waves
The solutions of Maxwell’s are wave-like, with both E and B satisfying a wave equation
Electromagnetic waves travel at the speed of light
This comes from the solution of Maxwell’s equations

22. Properties of EM Waves, 2
The components of the electric and magnetic fields of plane electromagnetic waves are perpendicular to each other and perpendicular to the direction of propagation
The electromagnetic waves are transverse waves

23. Properties of EM Waves, 3
The magnitudes of the fields in empty space are related by the expression
This also comes from the solution of the partial differentials obtained from Maxwell’s Equations
Electromagnetic waves obey the superposition principle

24. EM Wave Representation
This is a pictorial representation, at one instant, of a sinusoidal, linearly polarized plane wave moving in the x direction
E and B vary sinusoidally with x

25. Rays A ray is a line along which the wave travels
All the rays for the type of linearly polarized waves that have been discussed are parallel
The collection of waves is called a plane wave
A surface connecting points of equal phase on all waves, called the wave front, is a geometric plane

26. Doppler Effect for Light
Light exhibits a Doppler effect
Remember, the Doppler effect is an apparent change in frequency due to the motion of an observer or the source
Since there is no medium required for light waves, only the relative speed, v, between the source and the observer can be identified

27. Doppler Effect, cont.
The equation also depends on the laws of relativity
v is the relative speed between the source and the observer
c is the speed of light
ƒ’ is the apparent frequency of the light seen by the observer
ƒ is the frequency emitted by the source

28. Doppler Effect, final
For galaxies receding from the Earth, v is entered as a negative number
Therefore, ƒ’<ƒ and the apparent wavelength, l’, is greater than the actual wavelength
The light is shifted toward the red end of the spectrum
This is what is observed in the red shift