Maxwell s equations Physics /5/ Lecture XXIV Kyoto /5/ Lecture XXIV James Clerk Maxwell James Clerk Maxwell (83 879) Unification of electrical and magnetic interactions /5/ Lecture XXIV 3
Φ = da = Q ε Gauss s law the number of field lines that go in = the number of field lines that go out, unless there are sinks (negative charges) or sources (positive charges) /5/ Lecture IV 4 Gauss s law for magnetic field Can you guess: Φ = da =? /5/ Lecture XXIV 5 Gauss s law for magnetic field You can never separate north end of the magnet from south end à There are no magnetic charges (monopoles) - no sinks or sources à magnetic field lines are always closed à magnetic flux through any is always zero Φ = da = /5/ Lecture XXIV 6
Faraday s law of induction: Changing Φ generates Magnetic flux is increasing è induced emf: emf = dφ = V l = l = V lectric current in the circuit, but What makes stationary charges moves? Induced electric field: dφ Integral of the electric field over a is equal to negative derivative of the magnetic flux over time: /5/ Lecture XXI 7 dφ dl = Ampere s Law Current = moving electric charges produce magnetic field dl = µ I encl Integral of the magnetic field over a is equal to the enclosed current /5/ Lecture XII 8 Known phenomena Currents produce magnetic field Changing magnetic field produces electricity (induction) M theory Assumptions Changing electric field is equivalent to current and also produces magnetic field Unification James Clark Maxwell (83-879): one field electromagnetic (M). /5/ Lecture XXI 9 3
Displacement currents A Moving charged particles Q = CV = ( ε )( d) = ε A = ε Φ d (current) create magnetic field Charged particles create dq dφ " I D"= = ε electric field lectric field is changing when particles move Changing electric field is equivalent to current displacement current I D No charges jump across the capacitor à No charges are actually moving in the displacement current I I D /5/ Lecture XXI Modified Ampere s Law Current = moving electric charges + displacement current (changing electric field) produce magnetic field dl = µ ( I + I dl = µ I encl displ ) encl dφ + µ ε dq dφ " I D"= = ε /5/ Lecture XII Gauss s Law Maxwell s equations Gauss s law for magnetic field == there no magnetic charges == magnetic filed lines are always closed Faraday s law Changing magnetic field can create electric field Amper s law + extra term introduced by Maxwell displacement current Changing electric field can create magnetic field closed Q da = ε da = dφ dl = path dφ dl = µ I + µ ε /5/ Lecture XXIV 4
Maxwell s equations in vacuum No charges, no currents Changing magnetic field creates electric field Changing electric field creates magnetic field closed da = da = dφ dl = path dφ dl = µ ε /5/ Lecture XXIV 3 dφ dl = path M wave dφ dl = µ ε dl = ( + d) dy dy = ddy dφ d = dxdy d d = x dx dl = ddz dφ d = µ ε = µ ε dxdz t ε µ x y y d d = v = µ ε t dx t x 8 v = = = 3. m / s ε µ 8.85 4π closed 7 /5/ Lecture XXIV 4 M wave v v = sin( kx ϖt) = sin( kx ϖt) v = v x y z / = v π k = λ ϖ = πf ϖ fλ = = v k /5/ Lecture XXIV 5 5
nergy in M wave M waves transport energy nergy density: u = ε + µ Poynting vector (energy transported by M wave per unit time per unit area) S = µ Average energy per unit time per unit area S = rms rms µ /5/ Lecture XXIV 6 Intensity of waves nergy of oscillation U is proportional to amplitude squared U Intensity I, W/m energy / time power I = = area area Intensity I is proportional to amplitude squared, inversely proportional to r : I I r r /5/ Lecture XXIV 7 Average intensity lectric fields follows harmonic oscillation: = sin( ϖt) Intensity (brightness for light) I is proportional to electric field squared I Average over time (one period of oscillation) I: = I π T I = I T π sin ( ϖt) = I sin ϖt xdx = I π ϖt π /5/ Lecture XXIV 8 I = I sin ( ϖt) sin ( ϖt) dϖt = I ( cos x) dx = 6
λ M spectrum f c = fλ c speed of light (m/s) f frequency (Hz=/s) λ wavelength (m) /5/ Lecture XXIV 9 Radiation from an AC antenna Changing electric field creates magnetic field Changing magnetic field creates electric field Change propagates with a finite velocity lectromagnetic wave proof of unification /5/ Lecture XXIV Transmission and reception Antennas are used to transmit and to receive M waves Rod antennas transmit and receive component to rod Loop antennas component (use induction) loop /5/ Lecture XXIV 7
Modulations Amplitude modulation (AM) Frequency modulation (FM) /5/ Lecture XXIV 8