Q E ds = enclosed ε S 0 08 Fomulae Sheet 1 q 1q q Coulomb s law: F =. Electic field ceated by a chage q: E = 4πε 4πε Pemittivity of fee space: 0 1 = 9 10 4πε 0 9 Newton mete / coulomb = 9 10 9 0 N m Q ed Gauss s law (electic flux though a closed suface): E ds = enclos. ε Suface aea of a sphee of adius R is S = 4πR S 0 σ A jump of the electic field ove a chaged suface: δ E = σ ε0 Ehs Elhs = (when an axes is diected fom left to ight!) ε0 q Electic potential of a point chage q: V ( ) V ( ) =. Unit: 1volt=J/C 4πε 0 ε 0 = 8.85 10-1 C/(Vm) Definition of the electic potential diffeence: Ed = [ V ( ) V ( )] Consevation of enegy fo a chage Q: K + QV ( ) = const. Enegy of an electon in electic potential=1volt (electon volt): 1eV=1.6 10-19 J (J=1Joule). Q A Capacitance: C=Q/V; Paallel-plate capacito: C = = ε ε =K ε 0 ΔV d 1 1 1 1 RaRb Spheical capacito: = ; C = 4πε C 4πε Rinne Route Rb Ra Unit: 1F 1faad = 1 coulomb / volt 1pF 10 F 1 F 10 1 6 = μ = F Capacitos in paallel: C tot Capacito as enegy stoage = C C 1 + C + 3 + ε 0 = 8.85 10-1 F/m Capacitos in seies: Q CV U = = u=enegy density C +++++++++++++++++++++++++++++++++++++++++++++++++ ΔQ dq Definition of cuent I( t) = lim =. Unit: 1A=1ampee= C/s. Δ t 0 Δt dt Cuent I= qnvs ( q=chage, n=density, v=velocity, S the coss-section aea)... 1 1 / C 1 1 1 1 = + + C1 C C E () u () = ε C tot 3 +... Ohm s law: V/I=R; V=RI; V/R=I Resisto with a constant coss section: Resistivity ρ is measued in [Ωm]. Unit: 1Ω=V/A=Vs/C Length L R = ρ = ρ. coss sec tion ' s aea S
1 1 1 1 Resistos in seies R tot = R1 + R + R3 + Resistos in paallel = + + +... R tot R1 R R3 Similaity between esistance and capacitance: R 1/ C ρ 1/ ε Powe output (enegy loss ate): P= IV = RI = V / R. Unit: [J/s] qt () Dischaging capacito: qt ( ) = Qinitial exp( t/ RC) I = dq/ dt = ; RC negative I implies that the chage flows out fom the plate, i.e., it is dischaging Q final Chaging capacito qt ( ) = Qfinal[1 exp( t/ RC)] I() t = exp( t/ RC ) RC Kichhoff s ules: sum of the diected cuents in each of the junctions is zeo; sum of the voltage dops and ises along each of the closed loops is zeo. ++++++++++++++++++++++++++++++++++++++++++++++ Foce acting on a chage q moving in the magnetic field F qv B = Foce acting on an element dl of a cuent-caying conducto: F = Idl B Cycloton fequency: ω qb f = = π π m Dipoles. Electic dipole moment of a pai chages sepaated by d u u u ±q : p = qd ; u u Magnetic dipole moment u u of a u small u aea suounded by a cuent I: μ = I( ds) Toque [Nm]: τ = p E ; τ = μ B. u u u u Enegy of a dipole in a field: U = p E ; U = μ B μ0 qv ( ) Magnetic field ceated by a moving chage q (Biot-Savat law): B = 3 4π μ0 I( dl ) Magnetic field ceated by an element dl caying cuent I: db = 3 4π Units fo magnetic field 1T ( tesla) = 1N / C m / s = 1N / A m Pemeability μ π π π 7 7 7 0 = 4 10 T m/ A= 4 10 N s / C = 4 10 N / A μ Magnetic field ceated by a staight wie caying cuent I : 0 B = I π u Steady-state vesion of Ampee s law (cuent enclosed by a path): Bdl = μ I Magnetic field ceated by a solenoid: B = μ0ni, n=n/l is numbe of tuns pe unit length. Faaday s law (the EMF induced in a closed loop as esponse to a change of magnetic flux though the loop): dφ B E d = dt Ampee s law (including displacement cuent ceated by vaying in time electic fields): u dφ E Bd = μ0( Ic + ε0 ) dt contou 0 enclosed
Maxwell s equations: two Gauss s laws + Faaday s and Ampee s laws dφ B Mutual Inductance: Emf = N NΦ B = M I dt dφ B1 Emf1 = N1 N1Φ B1 = M1I dt Mutual Inductance: Mmutual = μ0nn 1 lovelapsovelap 1 1 M = M 1 1 flux [ Φ B] = 1T m = 1 N m s / C = 1 J s / C = 1V s Units fo flux (webe) and EMF: 1T m = 1Wb 1V = 1 Wb/ s Units of the mutual inductance (heny): 1heny = 1 H = 1 Wb /1A = 1 V s /1A = 1Ω s = 1 J / A Inductance (self-inductance): dφ = = dt B Emf N L 7 Anothe units fo pemeability: μ0 = 4π 10 H / m NΦ N Inductance of a tooidal solenoid: L π Emf Cuent gowth in an R-L cicuit: I = (1 exp( Rt / L)) R Decay of cuent in an R-L cicuit: I = It ( = 0)exp( Rt/ L) di dt = I B = μ 0 Aea () ( / ) Magnetic field enegy: Ut () = LI t = L dq dt B Density of magnetic field enegy ub = μ 1 q ( t) Q dq 1 () Oscillations in a L-C cicuit: + q = 0, ω = 1/ LC ; C C dt LC It () = Isin( ωt+ ϕ) 0 M Φ = I LI t + = = const ++++++++++++++++++++++++++++++++++++++++++++++ Waves (fequency, wave vecto, speed): π π λ ω = k = v= =ω/k ω =vk T λ T Wave popagating along x: y( x; t) ight / left = Acos( kx m ωt + ϕ) ϕ = phase Wave equation: ytx (, ) ytx (, ) v = x t Set of wave equations in electomagnetism: E B (, ) E (, ) z t x y t x εμ y(, t x) Bz (, tx) = = 0 0 x t x t Speed of light in vacuum and medium; index of efaction n: 1 1 c = = (3 10 m/ s ) 1 7 εμ (8.85 10 C / Nm ) (4π 10 N / A ) 8 0 0 v = n = c/ v = KKmagn K v = c/ n εμ 1 E (, t x) E (, t x) y y = εμ 0 0 x t B
Relation between the amplitudes of the electic and magnetic fields in electomagnetic fields: E=cB. Radiation powe: P=IA Intensity of adiation fa away fom the souce: I = P π /(4 ) Density of enegy: u = ε0 E ( x, t) ; aveage u u density of enegy u = ε0 E ( x, t) = ε0e / Poynting vecto S, intensity I: u E B u u EB S = P = S A I = S = μ0 μ0 Radiation pessue: P = αi / c; fo totally eflecting mio α=; fo black body α=1. ad +++++++++++++++++++++++++++++++++++++++++++++++++++ Angle of eflection: θ = θ Snell s law: n sinθ incident = n eflected sinθ incident incident efacted efacted n Angle of total intenal eflection: sinθ citical = n Polaizing by a linea filte along the diection Malus s law (consequence of the elation above): incident Huygens s and Femat s pinciples. efacted incident π π Polaizations: cicula E y = Ez ϕy = ϕz m ; elliptical E y Ez ϕy = ϕz m ; n linea : ϕy = ϕz. efacted Buste s angle: tg θ pola = n y ' s ' Images; lateal magnification: m = = y s 1 1 1 Concave spheical mio ), focal length: + = f = R / s s' f 1 1 1 Convex spheical mio ( : + = f = R / s s' f Spheical efactive image na nb nb na y' nas' + = m = = s s' R y n Thin lenses (conveging lens, f>0; diveging lens, f<0): 1 1 bs 1 s + = m = ' s s' f s 1 1 1 Lens make s equation: = ( n 1) Double convex/concave lenses: f R1 R 1 1 1 ( n 1) = + f R1 R u u filte n: Eincident n n E I = I cos φ max ( incident )
++++++++++++++++++++++++++++++++++++++++++++++ Integals: 1 N + 1 N x dx = x N + 1 R N 1 N + 1 x dx = R N + 1 0 1 1 1 dx N x N + 1 x = N 1 R 1 N x dx = 1 1 N 1 R N 1 1 dx = ln x x b 1 dx = ln( b / a) x a x 0 t d a dy ( ) 3/ x x + y x + a 0 1 1 = τ exp( τ / τ ) = τ (1 exp( t / τ ) a 0 0 0 Aveaging cos ( ωt kx) = sin ( ωt kx) = cos( ωt kx) sin( ωt kx) = 0 1 cos π /6= 3/ sin π /6= 1/ cos π / 3 = 1/ sin π / 3 = 3 /