2 Ampere’s Law is Wrong!Maxwell realized Ampere’s Law is not self-consistentThis isn’t an experimental argument, but a theoretical oneConsider a parallel plate capacitor getting charged by a wireConsider an Ampere surface between the platesConsider an Ampere surface in front of platesBut they must give the same answer!There must be something else that creates B-fieldsNote that for the first surface, there is also an electric field accumulating in capacitorMaybe electric fields?Take the time derivative of this formulaSpeculate : This replaces I for first surfaceIIThis is in Chapter 34 now. It used to be in chapter 30 and pg 37 of my notes.CT-5 from chapter 30 can now be used for 34.
3 Ampere’s Law (New Recipe) Is this self-consistent?Consider two surfaces with the same boundaryI1I2Gauss’s Law for electric fieldsE2E1This makes sense!B
5 All of electricity and magnetism is somewhere on this page Maxwell’s EquationsWe now have four formulas that describe how to get electric and magnetic fields from charges and currentsGauss’s LawGauss’s Law for MagnetismAmpere’s Law (final version)Faraday’s LawCollectively, these are called Maxwell’s EquationsThere is also a formula for forces on chargesCalled Lorentz ForceAll of electricity and magnetism is somewhere on this pageInside materials, these formulas are sometimes modified to include the effects of the reaction of the materials (like dielectric constants)These formulas commonly rewritten in derivative form
6 Electromagnetic Waves Wave solutionsWe can solve Maxwell’s Equations(take my word for it) and come up with two “simple” differential equations.I could have used cosine instead, it makes no differenceI chose arbitrarily to make it move in the x-directionThese are waves where we have .
8 Wave Equations summarized: Waves look like:Related by:Two independent solutions to these equations:E0B0Note that E, B, and direction of travel are all mutually perpendicularThe two solutions are called polarizationsWe describe polariza-tion by telling which way E-field pointsE0B0Do thing with y = x^2. y = (x-3)^2. y = (x-t)^2.Note there are other polarization (ex diagonal).Note E B is in direction of motion
9 Understanding Directions for Waves The wave can go in any direction you wantThe electric field must be perpendicular to the wave directionThe magnetic field is perpendicular to both of themRecall: E B is in direction of motionA wave has an electric field given by E = j E0 sin(kz – t). What does the magnetic field look like?A) B = i (E0/c) sin(kz - t) B) B = k (E0/c) sin(kz - t)C) B = - i (E0/c) sin(kz - t) D) B = - k (E0/c) sin(kz - t)The magnitude of the wave is B0 = E0 / cThe wave is traveling in the z-direction, because of sin(kz - t).The wave must be perpendicular to the E-field, so perpendicular to jThe wave must be perpendicular to direction of motion, to kIt must be in either +i direction or –i directionIf in +i direction, then E B would be in direction j i = - k, wrongSo it had better be in the –i direction
11 The meaning of c: Waves traveling at constant speed Keep track of where they vanishc is the velocity of these wavesThis is the speed of lightLight is electromagnetic waves!But there are also many other types of EM wavesThe constant c is one of the most important fundamental constants of the universe
12 Wavelength and wave number The quantity k is called the wave numberThe wave repeats in timeIt also repeats in spaceEM waves most commonly described in terms of frequency or wavelengthSome of these equations must be modified when inside a material
14 The Electromagnetic Spectrum Different types of waves are classified by their frequency (or wavelength)Boundaries are arbitrary and overlapVisible is nmRadio WavesMicrowavesInfraredVisibleUltravioletX-raysGamma Raysf Increasing IncreasingRedOrangeYellowGreenBlueVioletWhich of the following waves has the highest speed in vacuum?A) InfraredB) OrangeC) GreenD) BlueE) It’s a tieF) Not enough infoVermillionSaffronChartreuseTurquoiseIndigo
15 Energy and the Poynting Vector Let’s find the energy density in the waveNow let’s define the Poynting vector:It is energy density times the speed at which the wave is movingIt points in the direction energy is movingIt represents the flow of energy in a particular direction, energy/area/timeUnits:
16 Intensity and the Poynting vector The time-averaged Poynting vector is called the IntensityPower per unit areaIn Richard Williams’ lab, a laser can (briefly) produce 50 GW of power and be focused onto a region 1 m2 in area. How big are the electric and magnetic fields?
18 Momentum and Pressure Light carries energy – can it carry momentum? Yes – but it’s hard to provep is the total momentum of a wave and U the total energySuppose we have a wave, moving into a perfect absorber (black body)As they are absorbed, they transfer momentumIntensity:As waves hit the wall they transfer their momentumPressure on a perfect absorber:When a wave bounces off a mirror, the momentum is reversedThe change in momentum is doubledThe pressure is doubled
19 Cross-Section:To calculate the power falling on an object, all that matters is the light that hits itExample, a rectangle parallel to the light feels no pressureAsk yourself: what area does the light see?This is called the cross sectionIf light of intensity S hits an absorbing sphere of radius a, what is the force on that sphere?A) a2S/c B) 2a2S/c C) 4a2S/cAs viewed from any side, a sphere looks like a circle of radius aThe cross section for a sphere, then, is a2
20 Sample Problem:A 150 W bulb is burning at 6% efficiency. What is the force on a mirror square mirror 10 cm on a side 1 m away from the bulb perpendicular to the light hitting it?1 mLight is distributed in all directions equally over the sphere of radius 1 m
22 Sources of EM waves A charge at rest produces no EM waves There’s no magnetic fieldA charge moving at uniform velocity produces no EM wavesObvious if you were moving with the chargeAn accelerating charge produces electromagnetic wavesConsider a current that changes suddenlyCurrent stops – magnetic field diminishesChanging B-field produces E-fieldChanging E-field produces B-fieldYou have a wave+–
23 Simple AntennasTo produce long wavelength waves, easiest to use an antennaAC source plus two metal rodsSome charge accumulates on each rodThis creates an electric fieldThe charging involves a currentThis creates a magnetic fieldIt constantly reverses, creating a waveWorks best if each rod is ¼ of a wavelength longThe power in any direction is– – – – – –++++++distance r
24 Common Sources of EM Radiation Radio Waves AntennasMicrowaves Klystron, MagnetronInfrared Hot objectsVisible Outer electrons in atomsUltraviolet Inner electrons in atomsX-rays Accelerated electronsGamma Rays Nuclear reactions