Maxwell’s Equations (integral form)

1 NameEquationDescriptionGauss Law for ElectricityCharge and electric fieldsGauss Law for MagnetismMagnetic fieldsFaraday s LawElectrical effects from changing B fieldAmpere s LawMagnetic effects from current0= AdBrr0 QAdE= rrdtdldEB = rrMaxwell s Equations (integral form) idlB0 = rThere is a serious to be modified. + ?cylindersquare00= = AdBQAdEenclosedrrrr Gauss Law s works for ANYCLOSED SURFACE bagelsphereRemarks on Gauss Law s with different closed surfacesSurfaces forintegration of E fluxFish bowldtAdBdldE = rrrrFaraday s Law works for any closed Loop and ANYattached surface areadiskLine integraldefines the Closed loopThis is proven in Vector Calculus with Stoke s TheoremcylinderRemarks on Faraday s Law with different attached surfacesSurface areaintegration for B fluxGeneralized Ampere s Law and displacement currentAmpere s original law, , is the parallel plate capacitor and suppose a current icisflowing charging up the plate.

2 If Ampere s law is applied for thegiven path in either the plane surface or the bulging surface wewe should get the same results, but the bulging surface has ic=0,so something is missing. encloseIdlB0 = rGeneralized Ampere s Law and displacement currentMaxwell solved dilemma by adding an addition term called displacement current,iD= d E/dt,in analogy to Faraday s Law.() +=+= dtdiiidlBEcDc000 r()EEAEddACVq ==== cEidtddtdq= = Current is once more continuous: iDbetween the plates = iCin the wire. Summary of Faraday s LawdtdldEB = rrIf we form any closed loop, theline integral of the electric field equals the time rate change of magnetic flux through the surfaceenclosed by the loop. If there is a changing magnetic field, then there will beelectric fields induced in closed paths. The electric fieldsdirection will tend to reduce the changing B of Ampere s Generalized LawIf we form any closed loop, the line integral of the B field is nonzero ifthere is (constant or changing) current through the loop.

3 If there is a changing electric fieldthrough the loop, then there will be magnetic fields induced about a closed loop path. BE += dtdidlBEc00 rBCurrent icJames Clerk Maxwell (1831-1879) generalized Ampere s Law made Equations symmetric: a changing magnetic field produces an electric field a changing electric field produces a magnetic field Showed that Maxwell s Equations predicted electromagnetic waves and c =1/ 0 0 Unified electricity and magnetism and light. Maxwell s EquationsAll of electricity and magnetism can be summarized by Maxwell s Law for ElectricityCharge and electric fieldsGauss Law for MagnetismMagnetic fieldsFaraday s LawElectrical effects from changing B fieldAmpere s Law(modified by Maxwell)Magnetic effects from current andChanging E field0= AdBrr0 QAdE= rrdtdldEB = rrMaxwell s Equations (integral form) += dtdidlBEc00 rElectromagnetic Waves in free spaceA remarkable prediction of Maxwell s eqns is electric &magnetic fields can propagate in vacuum.

4 Examples of electromagnetic waves include; radio/TV waves,light, x-rays, and microwaves. Wireless, blue tooth, cell phones, s James Clerk Maxwell predicted radio - Heinrich Hertz demonstrated rapid variations ofelectric current could produce radio - Guglielmo Marconi sent and received his first radiosignal in Italy. JamesClerkMaxwellGuglielmoMarconiOn to Waves!! Note the symmetry now of Maxwell s Equations in free space, meaning when no charges or currents are present222221hhxvt =his the variable that is changing in space (x) and time (t). vis the velocity of the wave. Combining these Equations leads to wave Equations for EandB, , Do you remember the wave equation???220022xxEEzt =0= AdErr0= AdBrrdtdldEB = rrdtddlBE = 00 rReview of Waves from Physics 170222221hhxvt = The one-dimensional wave equation:)()(),(21vtxhvtxhtxh++ =has a general solution of the form:where h1represents a wave traveling in the +xdirection and h2represents a wave traveling in the -xdirection.

5 2=kkfv ==22fT ==()()tkxAtxh =cos,hx AA= amplitude = wavelengthf= frequencyv= speedk= wave number A specific solution for harmonic waves traveling in the +xdirection is:WavesTransverse Wave: The wave pattern moves to the right. However any particular point (look at the blue one) just moves transversely ( , up and down) to the direction of the Velocity: The wave velocity is defined as the wavelength divided by the time it takes a wavelength (green)to pass by a fixed point (blue).Velocity of Electromagnetic Waves We derived the wave equation forEx (Maxwell did it first, in ~1865!):220022xxEEzt = 10 m / svc == Comparing to the general wave equation: we have the velocity of electromagnetic waves in free space: This value is essentially identical to the speed of light measured by Foucault in 1860! Maxwell identified light as an electromagnetic =E& Bin Electromagnetic Wave Plane Harmonic Wave:)sin(0tkzEEx =)sin(0tkzBBy =kc= wherexzy00/Nothing special about (Ex, By); , could have (Ey, -Bx)BEc= Byis in phase with Exwhere are the unit vectors in the (E,B) directions.

6 ()be , bes = The direction of propagation is given by the cross products Lecture 21, ACT 3 Suppose the electric field in an e-m wave is given by: (a) + zdirection(b) -zdirectionWhich of the following expressions describes the magnetic field associated with this wave?3B0 cos()EyEkzt = +r(a) Bx= -(Eo/c) cos(kz+ t)(b) Bx= +(Eo/c) cos(kz- t)(c) Bx= +(Eo/c) sin(kz- t)3 AIn what direction is this wave traveling ?Lecture 21, ACT 3 Suppose the electric field in an e-m wave is given by: In what direction is this wave traveling ?(a) + zdirection(b) -zdirection3A0 cos()EyE kzt = +r To determine the direction, set phase = 0:0=+ tkz tkz += Therefore wave moves in +zdirection! Another way: Relative signs opposite means +directionLecture 21, ACT 3 Suppose the electric field in an e-m wave is given by: In what direction is this wave traveling ?

7 (a) + zdirection(b) -zdirection Which of the following expressions describes the magnetic fieldassociated with this wave?(a) Bx= -(Eo/c) cos(kz+ t)(b) Bx= +(Eo/c) cos(kz- t))(c) Bx= +(Eo/c) sin(kz- t)3A3B Bis in phase with Eand has direction determined from:esb = At t=0, z=0, Ey= -Eo Therefore at t=0, z=0, ijkesb ) ( = = =0 cos()EBikztc =+ r0 cos()EyE kzt = +rsinusoidal EM wave solutions; moving in +x()tkxEEy =cosmax()tkxBBz =cosmaxfk 22==kcf ==Properties of electromagnetic waves ( , light)Speed:in vacuum, always 3 108m/s, no matter how fast the source is moving (there is no aether !). In material, the speed can be reduced, usually only by ~ , but in 1999 to 17 m/s!In reality, light is often somewhat localized transversely ( , a laser) or spreading in a spherical wave ( , a star).A plane wave can often be a good approximation ( , the wavefronts hitting us from the sun are nearly flat).

8 Direction:The wave described by cos(kx- t) is traveling in the direction. This is a plane wave extends infinitely in and . +x y zPlane Wavesxzy For any given value of z, the magnitude of the electric field is uniform everywhere in the x-yplane with that zvalue.