1Maxwell’s equations and EM wavesThis LectureMore on Motional EMF and Faraday’s lawDisplacement currentsMaxwell’s equationsEM WavesFrom previous LectureTime dependent fields and Faraday’s LawRadar Remote Sensing of the AtmosphereStan BriczinskiApril 4, 20083Calculation of Electric/Magnetic Field for a moving charge•Coulomb's Law:€ dE =14πε0dqr2ur€ dB =µ04πIds × urr2• Biot-Savart+×INowcharge moves inside page4Gauss’ law in magnetostaticsNet magnetic flux through any closed surface is always zero (the number of lines that exit and enter the closed surface is equal)€ B • dA∫= 0No magnetic ‘charge’, so right-hand side=0 for B-fieldBasic magnetic element is the dipole€ E • dA∫=QenclosedεoCompare to Gauss’ law for E-fieldLorentz force in E and B-fieldsIf a charge moves in the presence of E-field and B-field 5 € r F = qr E + qr v ×r B velocity selectorIf we switch on an E-field parallel to B it will accelerate q and loops become more stretched as q gains speed6Summary on static E- and B-fieldsAmpere’s law true for static (no time dependence) electric fields. What happens to the Ampere’s law if fields depend on time?Gauss’ law Ampere’s law€ B • dA∫= 0€ E • dA∫=Qenclosedεo€ B • ds∫=µoIMagnetostaticsElectrostatics€ E • ds∫= ?0SurfaceSurfaceLineLine7Time-dependent B-fieldNot only charges produce E-fielda changing B-field also produces an E-field. This is not like a Coulomb field produced by a charge starting or stopping in the charge but it is a ‘circular’ E-field € E • ds∫= 0 becomes E • ds∫= 0 − dΦBdt€ ⇒ E ≠ −∇VFaraday’s lawRemember last lecture Faraday’s Law8€ ε= E • ds∫= −ddtΦB= −ddtB∫• dAMagnetic flux through surface bounded by pathIntegral over closed pathE is not conservative!!! € E • dsLenz’s lawThis ‘circular’ E-field is able to move charges around the loop around the changing B-fieldQuick QuizA rectangular loop of wire is pulled at a constant velocity from a region with B = 0 into a region of a uniform B-field. The induced current;A) will be zeroB) will be some constant value that is not zeroC) will increase linearly with time 9vL€ I =εR=dΦBdt= BLdxdt= BLvWhat happens when the loop exits the B-field region?IindxBind10Faraday’s and Lenz’s lawvSNB flux increases when magnet moves since B from N to SBind opposite to BB flux decreases when magnet moves, Bind parallel to B and I changes directionBBBindBindvSNMagnetic flux change: the magnitude of B changes in timeOther ways to change B-flux112) the area crossed by B lines changes with t (motional emf)3) The angle θ between B and normal to loop changes with t€ ΦB= BA cosθ12Quick Quiz: motional emfA conducting bar moves with velocity v. Which statement is true?A) + accumulates at the top, and - at the bottomB) no complete circuit, therefore, no charge accumulationC) - accumulates at the top, and + at the bottomTHERE IS AN ELECTRIC FIELD ACROSS THE BAR AND A POTENTIAL DIFFERENCE AT EQUILIBRIUM qE = qvB or E = vB Potential difference = v B LLv-++-vFB=-e vxB13Induced current in a moving rodEquivalentcircuit R resistance of circuit force due to induced current on bar (drag force opposing to bar motion)! Moving bar acts as a battery: € FB= I r l ×r B14Ampere’s laws with time-dep fieldsNot only charges produce E-field, a changing B-field produces a non conservative E-fielda changing E-field produces a B-field€ E • ds∫= 0 becomes E • ds∫= 0 − dΦBdt€ B • ds∫=µoI becomes B • ds∫=µoI +µoεodΦEdt€ ⇒ E ≠ −∇VFaraday’s lawAmpere-Maxwell’s law?Displacement current15Ampere’s law says we can considerany surface bounded by close line CBUT current through S1 is I ≠ 0and current through S2 I=0 Is there something wrong?There is an electric flux through S2A is the area of the capacitor platesCDisplacement current = conduction currentId is given by the variation of EI conduction current in wire€ ΦE= E • dA = E • dA =qε0∫S1+S2∫S2Ampere’s - Maxwell Law16€ B • ds∫=µoI becomes B • ds∫=µoI +µoεodΦEdtMagnetic fields are produced both by conduction currents and by time-varying electric fields€ +µ0IdTot current is conduction+ displacement17The 4 Maxwell’s Equations€ E ⋅ dA =qε0S∫ (Gauss' Law) B ⋅ dA = 0S∫ E ⋅ ds = −dΦBdt (Faraday - Henry) B ⋅ dsL∫=µ0I + L∫µ0ε0dΦEdt (Ampere - Maxwell law)Currents create a B-fieldA changing E-field can create a B-fieldcharged particles create an electric field An E-field can be created by a changing B-fieldConsequence: induced currentThere are no magnetic monopolesLorentz force€ F = q(E + v × B)18James Clerk Maxwell and the theory of electromagnetismElectricity and magnetism are deeply related1865 Maxwell: mathematical theory showing relationship between electric and magnetic phenomenaMaxwell’s equations predict existence of electromagnetic waves propagating through space at velocity of lightPermittivity of free space:Permeability of free space:= 2.99792458 x 108 m/s19Reminders: classification of wavesLongitudinal wave: medium particles move in direction parallel to direction of propagation of wave (eg. sound waves)Sound wave:sinusoidal variation of pressure in airA mechanical wave is a disturbance created by a vibrating object that travels through a medium from one location to anotherTransverse wave: medium particles move in direction perpendicular to direction of wave (waves on a pond, EM waves)20Reminders: wavesVelocity of waves: λ=vT ⇒ v = λ/T=λf21Reminders on wavesTraveling waves on a string along x obey the wave equation: y=wave functionGeneral solution : y(x,t) = f1(x-vt) + f2(x+vt) y = displacementTraveling wave: superposition of sinusoidal waves (produced by a source that oscillates with simple harmonic motion): y(x,t) = A sin(kx-ωt) y(x,t) = sin(kx+ ωt) A = amplitudek = 2π/λ = wave number λ = wavelength f = frequency T = 1/f = periodω = 2πf=2π/T angular frequencyλ€ ∂2y(x,t)∂x2=1v2∂2y(x,t)∂t2pulse traveling along +x pulse traveling along -xv22λFields exist even with q=0 and I=0!€ E ⋅ dA = 0S∫ (Gauss' Law) B ⋅ dA = 0S∫ E ⋅ ds = −dΦBdt (Faraday - Henry) B ⋅ dsL∫= L∫µ0ε0dΦEdt (Ampere - Maxwell law)Transverse wave composed of E and B
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