Chapter 35. Electromagnetic Fields and Waves

Transcription

1 Chapter 35. Electromagnetic Fields and Waves To understand a laser beam, we need to know how electric and magnetic fields change with time. Examples of timedependent electromagnetic phenomena include highspeed circuits, transmission lines, radar, and optical communications. Chapter Goal: To study the properties of electromagnetic fields and waves.

2 E or B? It Depends on Your Perspective Whether a field is seen as electric or magnetic depends on the motion of the reference frame relative to the sources of the field.

3 Transformations The Galilean field transformation equations are where V is the velocity of frame S' relative to frame S and where the fields are measured at the same point in space by experimenters at rest in each reference frame. NOTE: These equations are only valid if V << c.

4 Example transformation Consider a charge at rest at the origin in S where B=0 and E is given by Coulomb s Law. In S, the charge is moving and there is both an electric field E =E and a magnetic field B. Biot Savart Law

5 Maxwell s Equations and Electromagnetic Waves Maxwell s equations provide a unified description of the electromagnetic field and predict that Electromagnetic waves can exist at any frequency, not just at the frequencies of visible light. This prediction was the harbinger of radio waves. All electromagnetic waves travel in a vacuum with the same speed, a speed that we now call the speed of light.

6 Light speed Maxwell s equations predict EM waves move at a unique (light) speed relative to ANY frame of reference. That is impossible if the Galilean velocity addition rule v = v+v rel holds. This paradox is resolved by Einstein s Special Theory of Relativity.

7 Discovery of artificial EM waves by Hertz Transmitter spark gap Receiver spark gap EM wave Magnified view of the spark gap and dipole transmitting ("feed") antenna at the focal point of the reflector. The high voltage spark jumped the gap between the spherical electrodes. The electrical impulse produced by the spark generated damped oscillations in the dipole antenna. Magnified view of the spark gap and dipole receiving antenna at the focal point of a receiving reflector similar to the transmitting one. The width of the small spark gap on the right is controlled by the screw below it. The vertical dipole antenna at the left was about 40 centimeters long.

8 Schematic of operation of antenna An oscillating electric dipole is surrounded by an oscillating electric (and magnetic) field. Field disturbances propagate away at light speed perpendicular to the dipole. Operating in reverse, an EM wave excites charges oscillation in a dipole.

9 Marconi s transatlantic signal experiment Left to right: Kemp, Marconi, and Paget pose in front of a kite that was used to keep aloft the receiving aerial wire used in the transatlantic radio experiment.

10 Marconi s transatlantic signal generator Induction coils Capacitor banks Spark gap

11 Radio Like two identical tuning forks coupled resonantly through a sound wave, two tuned LC or other resonant circuits can be coupled through an EM wave. A high frequency EM can carry audio frequency analog information by warbling the frequency (FM) or amplitude (AM). An optimal antenna has a size of order a wavelength. More compact loop antennas work through induction.

12 Harmonic Waves A harmonic plane wave is generated by a single frequency source current distribution and is characterized by frequency f and wavelength c/f. E, B, v form a right handed coordinate system. A linearly polarized wave has the orientation of E fixed.

13 Generation of electromagnetic waves EM waves are emitted in general by accelerated charges which shed free force fields. Synchrotron light source Dipole antenna

14 EM waves are observed over a wide range of frequencies. Megahertz natural and artificial sources produce radio waves. Ultra high frequency motions inside nuclei source gamma ray radiation. EM wave spectrum

15 Properties of Electromagnetic Waves Any electromagnetic wave must satisfy four basic conditions: 1. The fields E and B and are perpendicular to the direction of propagation v em.thus an electromagnetic wave is a transverse wave. 2. E and B are perpendicular to each other in a manner such that E B is in the direction of v em. 3. The wave travels in vacuum at speed v em = c 4. E = cb at any point on the wave.

16 Energy of Electromagnetic Waves EM waves carry energy density u and momentum density u/c. Energy density in E-field Energy density in B-field u E = ε o E 2 ( r,t ) /2 u B = B 2 r,t ( ) /2µ o Total u Tot = ε o E 2 /2 + B 2 /2µ o = ε o E 2 /2 + E 2 /2c 2 µ o = ε o E 2 r,t ( ) = B 2 r,t ( ) /µ o u Tot = ε o E 2 = ε o E o 2 cos 2 kz ωt ( ) moves w/ EM wave at speed c

17 Poynting vector The energy flow of an electromagnetic wave is described by the Poynting vector defined as The magnitude of the Poynting vector is The intensity of an electromagnetic wave whose electric field amplitude is E 0 is

18 EXAMPLE 35.4 The electric field of a laser beam Laser light is monochromatic and sourced by coherent high frequency motion of atomic electrons.

19 EXAMPLE 35.4 The electric field of a laser beam

20 Radiation Pressure A electromagnetic wave carries momentum density U/c and if the momentum is absorbed or reflected a pressure is exerted called the radiation pressure p rad. The radiation pressure on an object that absorbs all the light is where I is the intensity of the light wave. Note reflection implies twice the momentum transfer and twice the pressure.

21 QUESTION: EXAMPLE 35.5 Solar sailing

22 EXAMPLE 35.5 Solar sailing

23 Plane Polarized x z y Polarization E = E o cos kz ωt ( ) x ˆ B = B o cos( kz ωt) y ˆ Unpolarized Superposition of plane polarized waves

24 Polarization filters An array of linear conductors absorbs energy only from the component of electric field along the conducting direction transmitting the orthogonal component.

25 E inc = ( E inc cosθ ) x ˆ + Polarization filters Plane-polarized incident wave E o cos kx ωt ( ) ( E inc sinθ) y ˆ y x polarizer transmitted absorbed Transmitted wave = E trans = E o cosθ cos( kx ωt) x ˆ transmission

26 Malus s Law Suppose a polarized light wave of intensity I 0 approaches a polarizing filter. θ is the angle between the incident plane of polarization and the polarizer axis. The transmitted intensity is given by Malus s Law: If the light incident on a polarizing filter is unpolarized, the transmitted intensity is In other words, a polarizing filter passes 50% of unpolarized light and blocks 50%.

27 Malus s Law Here we see the effect of two filters. When parallel, light transmitted by the first is transmitted by the second. When orthogonal, light transmitted by the first is absorbed by the second.

28 Unpolarized light reflected from a surface becomes partially polarized Polarization by reflection Degree of polarization depends on angle of incidence Unpolarized Incident light Reflection polarized with E-field parallel to surface n Refracted light

29 Glare reduction Reflected sunlight partially polarized. Horizontal reflective surface ->the E- field vector of reflected light has strong horizontal component. Transmission axis