Unit22_MaxwellsEquation

Maxwells EquationsThe two Gausss laws are symmetrical, apart from the absence of the term for magnetic monopoles in Gausss law for magnetismFaradays law and the Ampere-Maxwell law are symmetrical in that the line integrals of E and B around a closed path are related to the rate of change of the respective fluxes

Gausss law (electrical):The total electric flux through any closed surface equals the net charge inside that surface divided by eoThis relates an electric field to the charge distribution that creates it

Gausss law (magnetism): The total magnetic flux through any closed surface is zeroThis says the number of field lines that enter a closed volume must equal the number that leave that volumeThis implies the magnetic field lines cannot begin or end at any pointIsolated magnetic monopoles have not been observed in nature

Faradays law of Induction:This describes the creation of an electric field by a changing magnetic fluxThe law states that the emf, which is the line integral of the electric field around any closed path, equals the rate of change of the magnetic flux through any surface bounded by that pathOne consequence is the current induced in a conducting loop placed in a time-varying B

The Ampere-Maxwell law is a generalization of Amperes law

It describes the creation of a magnetic field by an electric field and electric currentsThe line integral of the magnetic field around any closed path is the given sum

Maxwells Equations in integral formGausss LawGausss Law for MagnetismFaradays LawAmperes Law

Maxwells Equations in free space (no charge or current)Gausss LawGausss Law for MagnetismFaradays LawAmperes Law

Hertzs ExperimentAn induction coil is connected to a transmitterThe transmitter consists of two spherical electrodes separated by a narrow gapThe discharge between the electrodes exhibits an oscillatory behavior at a very high frequencySparks were induced across the gap of the receiving electrodes when the frequency of the receiver was adjusted to match that of the transmitterIn a series of other experiments, Hertz also showed that the radiation generated by this equipment exhibited wave propertiesInterference, diffraction, reflection, refraction and polarizationHe also measured the speed of the radiation

Implication A magnetic field will be produced in empty space if there is a changing electric field. (correction to Ampere)This magnetic field will be changing. (originally there was none!)The changing magnetic field will produce an electric field. (Faraday)This changes the electric field.This produces a new magnetic field.This is a change in the magnetic field.

An antennaWe have changed the magnetic field near the antenna Hook up an AC sourceAn electric field results! This is the start of a radiation field.

Look at the cross sectionE and B are perpendicular (transverse) We say that the waves are polarized.E and B are in phase (peaks and zeros align)

Called:Electromagnetic WavesAccelerating electric charges give rise to electromagnetic waves

Angular Dependence of IntensityThis shows the angular dependence of the radiation intensity produced by a dipole antennaThe intensity and power radiated are a maximum in a plane that is perpendicular to the antenna and passing through its midpointThe intensity varies as (sin2 ) / r2

Harmonic Plane WavesxAt t = 0At x = 0ll = spatial period or wavelengthTT = temporal periodt

Applying Faraday to radiation

Applying Ampere to radiation

Fields are functions of both position (x) and time (t)Partial derivatives are appropriateThis is a wave equation!

The Trial SolutionThe simplest solution to the partial differential equations is a sinusoidal wave:E = Emax cos (kx t) B = Bmax cos (kx t)The angular wave number is k = 2/ is the wavelengthThe angular frequency is = 2 is the wave frequency

The trial solution

The speed of light (or any other electromagnetic radiation)

The electromagnetic spectrum

Another look

Energy in Waves

Poynting VectorPoynting vector points in the direction the wave movesPoynting vector gives the energy passing through a unit area in 1 sec.Units are Watts/m2

IntensityThe wave intensity, I, is the time average of S (the Poynting vector) over one or more cyclesWhen the average is taken, the time average of cos2(kx - t) = is involved

Radiation PressureMaxwell showed:(Absorption of radiation by an object)What if the radiation reflects off an object?

Pressure and MomentumFor a perfectly reflecting surface, p = 2U/c and P = 2S/cFor a surface with a reflectivity somewhere between a perfect reflector and a perfect absorber, the momentum delivered to the surface will be somewhere in between U/c and 2U/cFor direct sunlight, the radiation pressure is about 5 x 10-6 N/m2

*dS = n dAFlux = field integrated over a surfaceNo magnetiic monopolesE .dl is an EMF (volts)*dS = n dAFlux = field integrated over a surfaceNo magnetiic monopolesE .dl is an EMF (volts)