Maxwell’s Equations (integral Form)

Maxwell’s Equations (integral form)NameEquationGauss’ Law forElectricityr r Q E dA Gauss’ Law forMagnetismr r B dA 0ε0Faraday’s Lawr rdΦ B E dl dtAmpere’s Lawr B dl μ 0 iNeeds to be modified.There is a serious asymmetry.DescriptionCharge andelectric fieldsMagnetic fieldsElectrical effectsfrom changing BfieldMagnetic effectsfrom current ?

Remarks on Gauss Law’s with different closed surfacesr r Qenclosed E dA r r B dA 0ε0Gauss Law’s works forANY CLOSED SURFACEsquarecylinderSurfaces forintegrationof E fluxspherebagel

Remarks on Faraday’s Law with different attached surfacesr rr rd B dA E dl dtLine integraldefines theClosed loopdiskFaraday’s Law works forany closed Loop and ANYattached surface areaSurface areaintegration for B fluxcylinderFish bowlThis is proven in Vector Calculus with Stoke’s Theorem

Generalized Ampere’s Law and displacement currentr B dl μ0 I encloseAmpere’s original law,, is incomplete.Consider the parallel plate capacitor and suppose a current ic isflowing charging up the plate. 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.

Generalized Ampere’s Law and displacement currentMaxwell solved dilemma by adding an addition term calleddisplacement current, iD ε dΦE/dt, in analogy to Faraday’s Law.rdΦ E B dl μ 0 (ic iD ) μ 0 ic ε 0 dt Current is once more continuous: iD between the plates iC in the wire.q CVεA ( Ed )d εEA εΦ EdΦ Edq ε icdtdt

Summary of Faraday’s Lawr rdΦ B E dl dtIf we form any closed loop, theline integral of the electric fieldequals the time rate change ofmagnetic 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 field.BE

Summary of Ampere’s Generalized LawrdΦ E B dl μ0 ic ε 0dt Current icIf we form any closed loop, the lineintegral of the B field is nonzero ifthere is (constant or changing) currentthrough the loop.If there is a changing electric fieldthrough the loop, then there willbe magnetic fields induced about aclosed loop path.BEB

Maxwell’s EquationsJames 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 predictedelectromagnetic waves and c 1/ ε0μ0 Unified electricity and magnetism and light.All of electricity and magnetism can be summarized byMaxwell’s Equations.

Maxwell’s Equations (integral form)NameEquationDescriptionGauss’ Law forElectricityr r Q E dA Charge and electricfieldsGauss’ Law forMagnetismr r B dA 0ε0Faraday’s Lawr rdΦ B E dl dtAmpere’s Law(modified byMaxwell)rdΦ E B dl μ 0 ic ε 0dt Magnetic fieldsElectrical effectsfrom changing BfieldMagnetic effectsfrom current andChanging E field

Electromagnetic Waves in free spaceA remarkable prediction of Maxwell’s eqns is electric &magnetic fields can propagate in vacuum.Examples of electromagnetic waves include; radio/TV waves,light, x-rays, and microwaves. Wireless, blue tooth, cellphones, etc.1860’s – James Clerk Maxwell predicted radio waves.1886 - Heinrich Hertz demonstrated rapid variations ofelectric current could produce radio waves.1895 - Guglielmo Marconi sent and received his first radiosignal in Italy.JamesClerkMaxwellGuglielmoMarconi

On to Waves!! Note the symmetry now of Maxwell’s Equations in free space,meaning when no charges or currents are presentr r E dA 0r r B dA 0r rdΦ B E dl dtrdΦ E B dl μ 0ε 0 dt Combining these equations leads to wave equations forE and B, e.g., Ex Ex μ0ε 022 z t22 Do you remember the wave equation? 2h 1 2h 2 22 x v th is the variable that is changingin space (x) and time (t). v is thevelocity of the wave.

Review of Waves from Physics 170 h 1 h 2 22 x v t2 The one-dimensional wave equation:has a general solution of the form:2h( x, t ) h1 ( x vt ) h2 ( x vt )where h1 represents a wave traveling in the x direction andh2 represents a wave traveling in the -x direction. A specific solution for harmonic waves traveling in the xdirection is:h λh x , t A cos kx ω tA( )()2πω 2π f k λTωv λf 2πkxA amplitudeλ wavelengthf frequencyv speedk wave number

WavesTransverse Wave: The wave pattern moves tothe right. However any particular point(look at the blue one) justmoves transversely (i.e., upand down) to the direction ofthe wave.Wave Velocity: The wave velocity isdefined as the wavelengthdivided by the time it takes awavelength (green) to passby a fixed point (blue).

Velocity of Electromagnetic Waves We derived the wave equation forEx (Maxwell did it first, in 1865!): 2Ex 2Ex μ 0ε 02 z t2 Comparing to the general wave equation: 2 h1 2h 22 xv t2we have the velocity of electromagnetic waves in free space:1 3.00 10 8 m / s cv μ 0ε 0 This value is essentially identical to the speed of lightmeasured by Foucault in 1860!– Maxwell identified light as an electromagnetic wave.

E & B in Electromagnetic Wave Plane Harmonic Wave:E x E 0 sin( kz ω t )B y B0 sin( kz ω t )whereω kcxzy¾By is in phase with Ex¾B0 E0 / c¾The direction of propagation ŝis given by the cross product( )sˆ eˆ bˆwhere eˆ , bˆ are the unit vectors in the (E,B) directions.Nothing special about (Ex, By); e.g., could have (Ey, -Bx)

Lecture 21, ACT 3 Suppose the electric field in an e-m wave is given by:rE yˆ E0 cos( kz ω t )3A In what direction is this wave traveling ?(a) z direction(b) - z direction3B Which of the following expressions describes the magnetic fieldassociated with this wave?(a) Bx -(Eo/c) cos(kz ω t)(b) Bx (Eo/c) cos(kz -ω t)(c) Bx (Eo/c) sin(kz -ω t)

Lecture 21, ACT 3 Suppose the electric field in an e-m wave is given by:rE yˆ E 0 cos( kz ω t )3A – In what direction is this wave traveling ?(a) z direction(b) - z direction To determine the direction, set phase 0: kz ωt 0 Therefore wave moves in z direction! Another way: Relative signs opposite means directionz ωkt

Lecture 21, ACT 3 Suppose the electric field in an e-m wave is given by:rE yˆ E 0 cos( kz ω t )3A – In what direction is this wave traveling ?(a) z direction(b) - z direction3B Which of the following expressions describes the magnetic fieldassociated with this wave?(a) Bx -(Eo/c) cos(kz ω t)(b) Bx (Eo/c) cos(kz -ω t) )(c) Bx (Eo/c) sin(kz -ω t) B is in phase with E and has direction determined from: bˆ sˆ eˆ At t 0, z 0, Ey -Eo Therefore at t 0, z 0, bˆ sˆ eˆ kˆ ( ˆj) iˆrˆ E0B icc o s (k z ω t )

sinusoidal EM wave solutions; moving in xE y Emax cos(kx ωt )B z Bmax cos(kx ωt )k 2πλω 2πfλf c ωk

Properties of electromagnetic waves (e.g., light)Speed: in vacuum, always 3 108 m/s, no matter how fast thesource is moving (there is no “aether”!). In material, thespeed can be reduced, usually only by 1.5, but in 1999 to17 m/s!Direction: The wave described by cos(kx-ωt) is traveling inthe xˆ direction. This is a “plane” wave—extendsinfinitely in ŷ and ẑ .In reality, light is often somewhat localized transversely(e.g., a laser) or spreading in a spherical wave (e.g., astar).A plane wave can often be a good approximation (e.g., thewavefronts hitting us from the sun are nearly flat).

Plane Waves For any given value of z, the magnitude ofthe electric field is uniform everywhere inthe x-y plane with that z value.xzy

Note the symmetry now of Maxwell’s Equations in free space, meaning when no charges or currents are present 22 22 2 hh1 xv t h is the variable that is changing in space (x) and time (t). v is the velocity of the wave. Combining these equations leads to wave equations