Classical Theory of Electromagnetic Radiation Chapter 24 Classical Theory of Electromagnetic Radiation

Maxwell’s Equations Four equations (integral form) : Gauss’s law Gauss’s law for magnetism Faraday’s law Ampere-Maxwell law Add Lorentz to complete the list of fundamental equations of electricity and magnetism Lorentz eq – defines the meaning of electric and magnetic field in terms of their effect on charge + Lorentz force

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) very high and deep slab – fields only inside Is it consistent with Maxwell’s law? Note: strictly speaking fields don’t move, they just change in time

A Pulse and Gauss’s Laws Pulse is consistent with Gauss’s law very hign and deep slab – fields only insude Is it consistent with Maxwell’s law? Pulse is consistent with Gauss’s law for magnetism

A Pulse and Faraday’s Law Area does not move Since pulse is ‘moving’, B depends on time and thus causes E emf E=Bv Is direction right?

A Pulse and Ampere-Maxwell Law =0

A Pulse: Speed of Propagation E=Bv E=cB Spectacular result: Maxwell predicts speed of light for electromagnetic pulse in mid-1800 – but no one at this time could guess the electromagnetic nature of light, though the speed of light was already known. He got speed of light theoretically from constants epsilon and mu which are easy to measure in experiment Based on Maxwell’s equations, pulse must propagate at speed of light

Clicker In a time Dt, what is DFmag? A) 0; B) BvDt; C) BhvDt; D) Bxh; E) B(x+vDt)h

Clicker What is E? B 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? 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

Electromagnetic Radiation Last shown

Electromagnetic Spectrum Microwave: not because it is used in microwave, but because it has ~micron wavelength Visible – the whole spectrum from blue to red – the frequency barely changes twice!

Maxwell’s Theory of Electromagnetism (1831-1879) Light is electromagnetic wave! Challenge: Design an electric device which emits and detects electromagnetic waves 1865 Even result that electromagnetic pulses and waves can propogate in space was new and not well accepted. Maxwell’s theory was based on unusual mechanical ideas about the luminiferous ether and has not been universally accepted. Moreover, Michelson assisted by Morley, performed remakarbly clever experiment that proved the non-exsistence of this ether. In 1884 Maxwell eqs were redirived by Hertz without assumption of ether, these are the ones we know now as Maxwell’s eqs, Maxwell original eqs did not look like that (different form)

Accelerated Charges Electromagnetic pulse can propagate in space How can we initiate such a pulse? Short pulse of transverse electric field

Accelerated Charges Transverse pulse propagates at speed of light Since E(t) there must be B Direction of v is given by: E B v No magnetic field out of the circle Ordinary (steady) magnetic field out of page inside – same as radiative. But this is does not need to be the case – acceleration matters – if it moves up and then slows down – different direction.

Accelerated Charges: 3D

Magnitude of the Transverse Electric Field We can qualitatively predict the direction. What is the magnitude? Vectors a, r and E always in one plane Magnitude can be derived from Gauss’s law Field ~ -qa Derivation is too complex for this course Kink in the electric field Much slower than 1/r2 – makes it possible to affect matter that is very far from the accelerated charges – that is why we see very distant stars. 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 Derivation is too complex for this course Kink in the electric field Much slower than 1/r2 – makes it possible to affect matter that is very far from the accelerated charges – that is why we see very distant stars. Quali

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? 2 cm E=106 N/C B Erad a 1. Acceleration a=F/m=qE/m=1.78.1017 m/s2 2. The direction of the field is opposite to qa 3. The magnitude: E=1.44.10-7 N/C 4. The direction of propagation is given by What is the magnitude of the Coulomb field at the same location?

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 + C