29. Maxwell’s Equations & Electromagnetic Waves The Four Laws of Electromagnetism Ambiguity in Ampere’s Law Maxwell’s Equations Electromagnetic Waves Properties of Electromagnetic Waves The Electromagnetic Spectrum Producing Electromagetic Waves Energy & Momentum in Electromagetic Waves

How does a conversation travel between cell phones? By 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.

Example 29.1. A Capacitor A parallel plate capacitor with plate area A and spacing d is charging at the rate dV/dt. Show that the displacement current is equal to the current in the wires feeding the capacitor. Capacitor : (Current feeding capacitor) Displacement current : QED

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 :

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.

GOT IT? 29.1 At a particular point E of an EM wave points in the +y direction, while B points in the z direction. Is the propagation direction +x ; x ; either +x or x but you can’t tell which; y ; +z ; or not along any of the coordinate axes ?

Example 29.2. 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 .

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

GOT IT? 29.3 Lasers 1 & 2 emit light of the same color, & E in the beam from laser 1 is twice as strong as that in laser 2’s beam. How do their magnetic fields, intensities, and wavelengths compare? B1 = 2 B2 S1 = 4 S2 1 = 2

Example 29.3. 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.

Momentum & Radiation Pressure Maxwell : radiation momentum radiation pressure on absorbing surface radiation pressure on reflecting surface Cosmos 1, a solar light-sailing spacecraft, failed at launch in 2005.