Maxwell s equations Physics 122 01 14 2019 Lecture XXIV 1 Q E EdA 0 closed surface 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 01 14 2019 Lecture IV 2 Gauss s law for magnetic field Can you guess B B d A closed surface 01 14 2019 Lecture XXIV 3 Gauss s law for magnetic field You can never separate north end of the magnet from south end There are no magnetic charges monopoles no sinks or sources magnetic field lines are always closed magnetic flux through any closed surface is always zero B B d A 0 closed surface 01 14 2019 Lecture XXIV 4 Changing FB generates E Magnetic flux is increasing induced emf d B dt Electric current in the circuit but What makes stationary charges Integral of the electric field over a moves closed path is equal to negative Induced electric field derivative of the magnetic flux 1 d B over time E l 01 14 2019 l dt Lecture XXI d B Edl dt 5 Ampere s Law Current moving electric charges produce magnetic field B dl 0 I encl Integral of the magnetic field over a closed path is equal to the enclosed current 01 14 2019 Lecture XII 6 EM theory Known phenomena Currents produce Assumptions magnetic field Changing magnetic Changing electric field is equivalent to current field produces electricity induction and also produces magnetic field Unification James Clark Maxwell 1831 1879 one field electromagnetic EM 01 14 2019 Lecture XXI 7 Displacement currents Moving charged particles current create magnetic field Charged particles create electric field Electric field is changing when particles move Changing electric field is equivalent to current displacement current ID No charges jump across the capacitor No charges are actually moving in the displacement current 01 14 2019 A Q CV 0 Ed 0 AE 0 E d Lecture XXI dQ d E ID 0 dt dt I ID 8 Modified Ampere s Law Current moving electric charges displacement current changing electric field produce magnetic field B dl 0 I encl I displ dQ d E ID 0 dt dt d E B dl 0 I encl 0 0 dt 01 14 2019 Lecture XII 9 Maxwell s equations Gauss s Law Gauss s law for magnetic field there no magnetic charges magnetic filed lines are always closed Q EdA 0 closed surface BdA 0 closed surface 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 01 14 2019 d B Edl dt closed path d E Bdl 0 I 0 0 dt closed path Lecture XXIV 10 Maxwell s equations in vacuum E d A 0 No charges no currents closed surface BdA 0 closed surface Changing magnetic field creates electric field Changing electric field creates magnetic field 01 14 2019 d B Edl dt closed path d E Bdl 0 0 dt closed path Lecture XXIV 11 d B Edl dt closed path d E Bd l 0 0 dt closed path EM wave Edl closed path d B dt dE x dx Bdl dBdz closed path d E dE 0 0 0 0 dxdz dt dt dB dE 0 0 t dx dt 1 v 01 14 2019 E dE dy Edy dEdy 0 0 2 E 1 2 E 2 t 0 0 x 2 1 8 85 10 12 4 10 7 Lecture XXIV dB dxdy dt dB dt 2 2 y 2 y v 2 t x 2 3 0 108 m s 12 EM wave E B E v B v 01 14 2019 E y E0 sin kx t Bz B0 sin kx t v x v E B v Lecture XXIV 2 k 2 f f v k 13 Energy in EM wave EM waves transport energy Energy density 1 1 2 2 u 0E B 2 2 0 Poynting vector energy transported by EM wave per unit time per unit area 1 S 0 E B Average energy per unit time per unit area 01 14 2019 1 S Erms Brms 0 Lecture XXIV 14 Intensity of waves Energy of oscillation U is proportional to amplitude squared E2 U E2 Intensity I W m2 energy time power I area area Intensity I is proportional to amplitude squared E2 inversely proportional to r2 I E 01 14 2019 2 Lecture XXIV 1 I 2 r 1 E r 15 Average intensity Electric fields E follows harmonic oscillation E E0 sin t Intensity brightness for light I is proportional to electric field squared 2 2 I E I I 0 sin t Average over time one period of oscillation I T 1 1 2 I I 0 sin t dt I 0 T 0 T 1 I 0 2 2 T 2 sin t d t 0 2 I0 1 1 2 sin xdx I 0 1 cos 2 x dx 2 2 0 2 0 01 14 2019 Lecture XXIV 16 EM spectrum f c f c speed of light m s f frequency Hz 1 s l wavelength m 01 14 2019 Lecture XXIV 17 Radiation from an AC antenna Changing electric field creates magnetic field Changing magnetic field creates electric field Change propagates with a finite velocity Electromagnetic wave proof of unification 01 14 2019 Lecture XXIV 18 Transmission and reception Antennas are used to transmit and to receive EM waves Rod antennas transmit and receive E component E to rod Loop antennas B component use induction B loop 01 14 2019 Lecture XXIV 19 Modulations Amplitude modulation AM Frequency modulation FM 01 14 2019 Lecture XXIV 20