# Short Version : 29. Maxwell’s Equations & EM Waves

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Short Version : 29. Maxwell’s Equations & EM Waves

29.1. The Four Laws of Electromagnetism
4 Laws of EM (incomplete) Law Mathematical Statement Physical Meaning How q produces E; E lines begin & end on q’s. Gauss for E No magnetic monopole; B lines form loops. Gauss for B Changing B gives emf. Faraday Ampere (Steady I only) Moving charges give B. Note E-B asymmetry between the Faraday & Ampere laws.

29.2. Ambiguity in Ampere’s Law
B in a RC circuit. Ampere’s law: I is current through any open surface S bounded by C. Current flows through surfaces 1,2,& 4. But not 3.  Ampere’s law fails ( for non-steady current ). Maxwell’s modification: Changing E gives I , which in turn gives B.

29.3. Maxwell’s Equations Law Mathematical Statement Physical Meaning
How q produces E; E lines begin & end on q’s. Gauss for E Gauss for B No magnetic monopole; B lines form loops. Faraday Changing B gives emf. Moving charges & changing E give B. Ampere-Maxwell Maxwell’s Eqs (1864). Classical electromagnetism.

Maxwell’s Equations in Vacuum
Gauss for E Gauss for B Faraday Ampere-Maxwell

29.4. Electromagnetic Waves
Faraday’s law: Ampere-Maxwell’s law: changing B gives E. changing E gives B.  Electromagnetic (EM) waves

Plane Electromagnetic Wave
EM wave in vacuum is transverse: E  B  k (direction of propagation). Right-hand rule For uniqueness, see Prob 46 Sinusoidal plane waves going in x-direction:

Gauss’s Laws Plane wave : Both E & B field lines are straightlines,
so their flux over any closed surfaces vanish identically. Hence the Gauss’s laws are satisfied.

Faraday’s Law For loop at x of height h & width dx : Faraday’s Law :
Faraday’s law expressed as a differential eq :

Ampere-Maxwell Law I = 0 For loop at x of height h & width dx :
Ampere-Maxwell law expressed as a differential eq : in vacuum

Conditions on Wave Fields
For E = E(x,t) j & B = B(x,t) k, Faraday’s Law : Ampere-Maxwell Law : For a plane wave Faraday’s Law : Ampere-Maxwell Law :

29.5. Properties of Electromagnetic Waves
speed of wave = = speed of light in vacuum = c Maxwell: light is EM wave. 1983: meter is defined so that c is exactly 299,792,458 m/s. Hence, 0 = 1 / (4  c2  107 ) C2/Nm2, where c = 299,792,458.

Example Laser Light A laser beam with wavelength 633 nm is propagating through air in the +z direction. Its electric field is parallel to the x axis and has magnitude 6.0 kV/m. Find the wave frequency, the amplitude of the magnetic field, and the direction of the magnetic field. (a) (b) (c) y axis.

Polarization Polarization  // E.
Radiation from antennas are polarized. E.g., radio, TV, …. Light from hot sources are unpolarized. E.g., sun, light bulb, … Reflection from surfaces polarizes. E.g., light reflecting off car hoods is partially polarized in horizontal direction. Transmission through crystal / some plastics polarizes. E.g., Polaroid sunglasses, … Only component of E // preferred direction e is transmitted.  = angle between Einc & . Law of Malus : or

Liquid crystal display (LCD)
2 polarizers with mutually perpendicular transmission axes. No light gets through where they overlap. Polarization of EM wave gives info about its source & the medium it passes through. Applications: astronomy, geological survey, material stress analysis, … Liquid crystal display (LCD) Unpolarized light Horizontal polarizer passes light. Horizontal polarizer blocks light. Applied V aligns molecules; polarization not rotated. LC molecules rotate polarization to horizontal direction. Vertical polarizer passes only Ev .

Conceptual Example 29.1. Crossed Polarizers
Unpolarized light shines on a pair of polarizers with their transmission axes perpendicular, so no light gets through the combination. What happens when a third polarizer is sandwiched in between, with its transmission axes at 45 to the others? 1st & middle polarizers not  so some light passes through. Passed light’s polarization not  to axis of last polarizer so some light passes through.

Making the Connection How does the intensity of light emerging from this polarizer “sandwich” compare with the intensity of the incident unpolarized light? Intensity of light emerging from 1st polarizer : ( polarized along axis of 1st polarizer ) Intensity of light emerging from middle polarizer : ( polarized along axis of middle polarizer.) Intensity of light emerging from ensemble : ( polarized along axis of 3rd polarizer.)

29.6. The Electromagnetic Spectrum
Earth’s atmosphere: Transparent to: most radio, visible light. Opaque to: most IR, upper UV, X-rays,  rays. UV is absorbed by ozone layer IR by green house gases.

29.7. Producing Electromagetic Waves
Any changing E or B will create EM waves. Any accelerated charge produces radiation. Radio transmitter: e’s oscillate in antenna driven by LC circuit. X-ray tube: accelerated e’s slammed into target. MW magnetron tube: e’s circle in B. EM wave : f = f of q motion Most efficient:  ~ dimension of emitter / reciever Outgoing EM waves Source replenishes radiated energy LC oscillator drives I in antenna Waves emit / receive  axis of dipole.

29.8. Energy & Momentum in Electromagetic Waves
Consider box of thickness dx, & face A  k of EM wave. Energy densities: Energy in box: Rate of energy moving through box: Intensity S = rate of energy flow per unit area Plane waves:

Plane waves: In general: see Prob 64 Poynting vector Average intensity for plane waves : E, B in phase

Example Solar Energy The average intensity of noontime sunlight on a clear day is about 1 kW/m2. What are the peak electric & magnetic fields in sunlight ? At this intensity, what area of 40% efficient solar collectors would you need to replace a 4.8-kW water heater ? (a) (b) Area needed is

Waves from Localized Sources
Afar from localized source, wave is spherical : Intensity = power / area  wave fields dominates static fields away from the sources.

Example 29.4. Cell Phone Reception
A cell phone’s typical average power output is about 0.6 W. If the receiver at a cell tower can handle signals with peak electric fields as weak as 1.2 mV/m, what is the maximum allowable distance from cell phone to tower ?

Application: Cell Phones
Hexagonal cell area  25 km2. ~ circle of radius Transmission & reception are at different frequencies.