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Chapter 24 Classical Theory of Electromagnetic Radiation Maxwell’s Equations Four equations (integral form) : Gauss’s law E nˆdA q inside 0 Gauss’s law for magnetism Faraday’s law Ampere-Maxwell law + Lorentz force d E dl dt B nˆdA d elec B dl 0 I inside_ path 0 dt F qE qv B Fields Without Charges Time varying magnetic field makes electric field Time varying electric field makes magnetic field Do we need any charges around to sustain the fields? Is it possible to create such a time varying field configuration which is consistent with Maxwell’s equation? Solution plan: • Propose particular configuration • Check if it is consistent with Maxwell’s eqs • Show the way to produce such field • Identify the effects such field will have on matter • Analyze phenomena involving such fields A Simple Configuration of Traveling Fields Key idea: Fields travel in space at certain speed Disturbance moving in space – a wave? 1. Simplest case: a pulse (moving slab) Note: strictly speaking fields don’t move, they just change in time A Pulse and Gauss’s Laws E nˆdA q inside 0 E nˆdA 0 Pulse is consistent with Gauss’s law B nˆA 0 Pulse is consistent with Gauss’s law for magnetism A Pulse and Faraday’s Law emf d mag dt Since pulse is ‘moving’, B depends on time and thus causes E mag Bhv t mag d mag Bhv t dt emf emf E dl Eh Is direction right? Area does not move E=Bv A Pulse and Ampere-Maxwell Law =0 d elec B dl 0 I inside_ path 0 dt elec Ehvt elec d elec Ehv t dt B dl Bh Bh 0 0 Evh B 0 0vE A Pulse: Speed of Propagation B 0 0vE E=Bv B 0 0vBv 1 0 0v 2 v 1 0 0 3 108 m/s E=cB Based on Maxwell’s equations, pulse must propagate at speed of light Clicker In a time t, what is mag? A) 0; B) Bvt; C) Bhvt; D) Bxh; E) B(x+vt)h Clicker d mag emf Bvh dt r r — Egdl Eh What is E? A) Bvh; B) Bv; C) Bvh/(2h+2x); D) B; E) Bvh/x Exercise If the magnetic field in a particular pulse has a magnitude of 1x10-5 tesla (comparable to the Earth’s magnetic field), what is the magnitude of the associated electric field? E cB E 3x108 m / s 1x105 T 3000V / m Force on charge q moving with velocity v perpendicular to B: Fmag/Fele = qvB/qE = vB/cB=v/c Direction of Propagation Direction of speed is given by vector product EB Electromagnetic Radiation Electromagnetic Spectrum Maxwell’s Theory of Electromagnetism • Light is electromagnetic wave! • Challenge: Design an electric device which emits and detects electromagnetic waves (1831-1879) Accelerated Charges Electromagnetic pulse can propagate in space How can we initiate such a pulse? Short pulse of transverse electric field Accelerated Charges 1. Transverse pulse propagates at speed of light 2. Since E(t) there must be B 3. Direction of v is given by: E B E v B Accelerated Charges: 3D Magnitude of the Transverse Electric Field We can qualitatively predict the direction. What is the magnitude? Magnitude can be derived from Gauss’s law Vectors a, r and E always in one plane Field ~ -qa Eradiative 1 qa 40 c 2 r 1. The direction of the field is opposite to qa 2. The electric field falls off at a rate 1/r Exercise a An electron is briefly accelerated in the direction shown. Draw the electric and magnetic vectors of radiative field. E B 1. The direction of the field is opposite to qa 2. The direction of propagation is given by E B Exercise An electric field of 106 N/C acts on an electron for a short time. What is the magnitude of electric field observed 2 cm away? 1. Acceleration a=F/m=qE/m=1.78.1017 m/s2 E=106 N/C B 2 cm a 2. The direction of the field is opposite to qa 3. The magnitude: 1 qa .10-7 N/C E=1.44 Eradiative 40 c 2 r 4. The direction of propagation is given by E B Erad What is the magnitude of the Coulomb field at the same location? q 6 E 3 . 6 10 N/C 2 40 r 1 Question A proton is briefly accelerated as shown below. What is the direction of the radiative electric field that will be detected at location A? B A A + D C