Lecture 22: TUE 13 APR 2010 Ch.32.1–5: Maxwell’s equations Ch.33.1–3: Electromagnetic Waves James Clerk Maxwell ( ) Physics 2102 Jonathan Dowling

EXAM 03: 6PM THU 15 APR LOCKETT 6 The exam will cover: Ch.28 (second half) through Ch (displacement current, and Maxwell's equations). The exam will be based on: HW07 – HW10. The formula sheet for the exam can be found here: You can see examples of old exam IIIs here:

Maxwell I: Gauss’ Law for E-Fields: Maxwell I: Gauss’ Law for E-Fields: charges produce electric fields, field lines start and end in charges S S SS

Maxwell II: Gauss’ law for B-Fields: Maxwell II: Gauss’ law for B-Fields: field lines are closed or, there are no magnetic monopoles S SS S SS

Maxwell III: Ampere’s law: Maxwell III: Ampere’s law: electric currents produce magnetic fields C

Maxwell IV: Faraday’s law: Maxwell IV: Faraday’s law: changing magnetic fields produce (“induce”) electric fields

Maxwell Equations I – IV:

In Empty Space with No Charge or Current …very suspicious… NO SYMMETRY! ? q=0 i=0

Maxwell’s Displacement Current If we are charging a capacitor, there is a current left and right of the capacitor. Thus, there is the same magnetic field right and left of the capacitor, with circular lines around the wires. But no magnetic field inside the capacitor? With a compass, we can verify there is indeed a magnetic field, equal to the field elsewhere. But there is no current producing it! ? E BB The missing Maxwell Equation!

Maxwell’s Fix We can write the current as: We calculate the magnetic field produced by the currents at left and at right using Ampere’s law : q=CV V=Ed C=0A/dC=0A/d  E =  EdA=EA E i d =  0 d  /dt

B ! E i B i B Displacement Current Maxwell proposed it based on symmetry and math — no experiment!

Maxwell’s Equations I – V: I II III IV V

Maxwell Equations in Empty Space: Fields without sources? Changing E gives B. Changing B gives E.

A solution to the Maxwell equations in empty space is a “traveling wave”… The electric-magnetic waves travel at the speed of light? Light itself is a wave of electricity and magnetism! Maxwell, Waves, and Light electric and magnetic fields can travel in EMPTY SPACE!

First person to use electromagnetic waves for communications: Guglielmo Marconi ( ), 1909 Nobel Prize (first transatlantic commercial wireless service, Nova Scotia, 1909) Electromagnetic waves First person to prove that electromagnetic waves existed: Heinrich Hertz ( )

Electromagnetic Waves: One Velocity, Many Wavelengths! with frequencies measured in “Hertz” (cycles per second) and wavelength in meters.

How do E&M Waves Travel? Is there an “ether” they ride on? Michelson and Morley looked and looked, and decided it wasn’t there. How do waves travel??? Electricity and magnetism are “relative”: Whether charges move or not depends on which frame we use… This was how Einstein began thinking about his “theory of special relativity”… We’ll leave that theory for later.

A solution to Maxwell’s equations in free space: Visible light, infrared, ultraviolet, radio waves, X rays, Gamma rays are all electromagnetic waves. Electromagnetic Waves

Radio waves are reflected by the layer of the Earth’s atmosphere called the ionosphere. This allows for transmission between two points which are far from each other on the globe, despite the curvature of the earth. Marconi’s experiment discovered the ionosphere! Experts thought he was crazy and this would never work.

Fig The wavelength/frequency range in which electromagnetic (EM) waves (light) are visible is only a tiny fraction of the entire electromagnetic spectrum. Maxwell’s Rainbow Fig (33-2)

An LC oscillator causes currents to flow sinusoidally, which in turn produces oscillating electric and magnetic fields, which then propagate through space as EM waves. Fig Oscillation Frequency: Next slide The Traveling Electromagnetic (EM) Wave, Qualitatively (33-3)

Fig Mathematical Description of Traveling EM Waves Electric Field:Magnetic Field: Wave Speed: Wavenumber: Angular frequency: Vacuum Permittivity: Vacuum Permeability: All EM waves travel a c in vacuum Amplitude Ratio: Magnitude Ratio: EM Wave Simulation (33-5)

Electromagnetic waves are able to transport energy from transmitter to receiver (example: from the Sun to our skin). The power transported by the wave and its direction is quantified by the Poynting vector. John Henry Poynting ( ) The Poynting Vector: Points in Direction of Power Flow E B S Units: Watt/m 2 For a wave, since E is perpendicular to B: In a wave, the fields change with time. Therefore the Poynting vector changes too!! The direction is constant, but the magnitude changes from 0 to a maximum value.

The average of sin 2 over one cycle is ½: Both fields have the same energy density. or, EM Wave Intensity, Energy Density A better measure of the amount of energy in an EM wave is obtained by averaging the Poynting vector over one wave cycle. The resulting quantity is called intensity. Units are also Watts/m 2. The total EM energy density is then

Solar Energy The light from the sun has an intensity of about 1kW/m 2. What would be the total power incident on a roof of dimensions 8m x 20m ? I = 1kW/m 2 is power per unit area. P=IA=(10 3 W/m 2 ) x 8m x 20m=0.16 MegaWatt!! The solar panel shown (BP- 275) has dimensions 47in x 29in. The incident power is then 880 W. The actual solar panel delivers 75W (4.45A at 17V): less than 10% efficiency…. The electric meter on a solar home runs backwards — Entergy Pays YOU!

The intensity of a wave is power per unit area. If one has a source that emits isotropically (equally in all directions) the power emitted by the source pierces a larger and larger sphere as the wave travels outwards: 1/r 2 Law! So the power per unit area decreases as the inverse of distance squared. EM Spherical Waves

Example A radio station transmits a 10 kW signal at a frequency of 100 MHz. At a distance of 1km from the antenna, find the amplitude of the electric and magnetic field strengths, and the energy incident normally on a square plate of side 10cm in 5 minutes. Received energy: