208 Formulae Sheet r 1 q1 q2 r q Coulomb’s law: F = 2 . Electric field created by a charge q: E = 2 4πε0 r 4πε 0r 1 Permittivity of free space: =9 × 109 Newton ⋅meter 2 / coulomb2 =9 ⋅ 109N ⋅ m 2 / C 2 4πε0 r r Q E⋅ dS = enclosed ∫ ε S 0 r r Q Gauss’s law (electric flux through a closed surface): E ⋅ d S = enclos ed . ∫ ε Surface area of a sphere of radius R is SR= 4π 2 S 0 r ||σ A jump of the electric field over a charged surface: δ ||E = σ ε0 EE−= (when an axes is directed from left to right !) rhs lhs ε 0 q Electric potential of a point charge q: V() r− V ( ∞ ) = . Unit: 1volt=J/C 4πε 0r -12 ε0 = 8.85×10 C/(Vm) r r2 r Edrr =−[() V rrr − V ()] r Definition of the electric potential difference: ∫ 21 r r1 Conservation of energy for a charge Q: K+ QV() r= const . Energy of an electron in electric potential=1volt (electron volt): 1eV=1.6×10-19 J (J=1Joule). Q A Capacitance: C=Q/V; Parallel-plate capacitor: C = = ε ε =K ε0 ΔV d 11⎛⎞ 1 1 RabR Spherical capacitor: =−⎜⎟; C = 4πε CRR4πε ⎝⎠inner outer Rba− R Unit: 11F≡= farad 1 coulomb / volt −− 12 6 110110pFFF==μ F -12 ε0 = 8.85×10 F/m 1 1 1 1 Capacitors in parallel: C = CCC + + + ... Capacitors in series: = + + + ... tot 1 2 3 CCCC tot 1 2 3 r QCV22 r E 2 ()r Capacitor as energy storage U == u=energy density ur()= ε 22C 2 +++++++++++++++++++++++++++++++++++++++++++++++++ ΔQ dQ Definition of current I( t )= lim = . Unit: 1A=1ampere= C/s. Δt →0 Δt dt Current I= qnvS ( q=charge, n=density, v=velocity, S the cross-section area) Ohm’s law: V/I=R; V=RI; V/R=I Unit: 1Ω=V/A=Vs/C Length L Resistor with a constant cross section: R ==ρ ρ . cross− sec tion ' s area S Resistivity ρ is measured in [Ωm]. 1 1 1 1 Resistors in series RRRRtot =1 + 2+ 3 +… Resistors in parallel = + + + ... RRRRtot 1 2 3 Similarity between resistance and capacitance: RC⇔ 1/ρ ⇔ 1/ε Power output (energy loss rate): PIVRIVR==22 =/ . Unit: [J/s] qt() Discharging capacitor: qt( )=− Q exp( t / RC ) Idqdt==−/ ; initial RC negative I implies that the charge flows out from the plate, i.e., it is discharging Q Charging capacitor qt( )=−− Q [1 exp( t / RC )] I()tt=−final exp( /RC ) final RC Kirchhoff’s rules: sum of the directed currents in each of the junctions is zero; sum of the voltage drops and rises along each of the closed loops is zero. ++++++++++++++++++++++++++++++++++++++++++++++ r r Force acting on a charge q moving in the magnetic field F =qvr × B r r Force acting on an element dl of a current-carrying conductor: F= Idl × B ω qB Cyclotron frequency: f == 22π πm ur ur ur Dipoles. Electric dipole moment of a pair charges ± q separated by d : p = qd ; ur ur Magnetic dipole moment of a small area surrounded by a current I: μ = I()dS rururrurur Torque [Nm]: τ =×p E ; τμ=×B . urur urur Energy of a dipole in a field: Up= −⋅E; UB= −⋅μ r r r μ0 qv()× r Magnetic field created by a moving charge q (Biot-Savart law): B = 3 4π r r r r μ0 I()dl× r Magnetic field created by an element dl carrying current I: dB = 3 4π r Units for magnetic field 1T ( tesla )= 1 N / C⋅ m / s= 1 N / A⋅ m −−7722−72 Permeability μπ0 =×410/TmA ⋅ =× 410π Ns ⋅ / C =× 410/π N A μ I Magnetic field created by a straight wire carrying current I : ||B = 0 2π r Steady-state version of Ampere’s law (current enclosed by a path): ur r ∫ Bdl= μ0 Ienclosed contour Magnetic field created by a solenoid: B = μ0nI , n=N/l is number of turns per unit length. Faraday’s law (the EMF induced in a closed loop as response to a change of magnetic flux r dΦ through the loop): E⋅ drr = − B ∫ dt Ampere’s law (including “displacement” current created by varying in time electric fields): urr dΦ E ∫ Bdr=+με00() Ic dt Maxwell’s equations: two Gauss’s laws + Faraday’s and Ampere’s laws dΦ Mutual Inductance: Emf=− NB2 NΦ = M I 22dt 222B 11 Φ M = M M = B dΦ 21 12 Emf=− NB1 NΦ = M I I 11dt 1112B 2 Mutual Inductance: MnnlSmutual = μ012overlap overlap flux[]1Φ =⋅=⋅⋅=⋅=⋅ T m2 1 N m s /1/1 C J s C V s Units for flux (weber) and EMF: B 1111/Tm⋅=2 WbV = Wbs Units of the mutual inductance (henry): 11henry== H1/11/111/ Wb A =⋅ V s A =Ω⋅= s J A2 ddΦ I Inductance (self-inductance): Emf=− NB =− L dt dt −7 Another units for permeability: μπ0 =×410/Hm NNΦ 2 Inductance of a toroidal solenoid: LA==B μ ×rea Ir0 2π Emf Current growth in an R-L circuit: I =−−(1 exp(Rt / L )) R Decay of current in an R-L circuit: I = It(=− 0)exp( RtL / ) LI22() t ( dQ / dt ) Magnetic field energy: Ut()==L 22 B2 Density of magnetic field energy uB = 2μ0 22 2 1()2 qt Q dq 1 2 LIt()+==const Oscillations in a L-C circuit: + q = 0 , ω = 1/LC ; 222CC dt2 LC It()=+ I sin(ωϕ t ) ++++++++++++++++++++++++++++++++++++++++++++++ Waves (frequency, wave vector, speed): 2π 2πλ ω =k=v==ω/k ω =vk T λ T Wave propagating along x: y(;) x tright/ left = A cos( kxm ω t+=ϕϕ ) phase Wave equation: 22 2 ∂∂ytx(, ) ytx (, ) v 22= ∂∂xt Set of wave equations in electromagnetism: 22 ∂Etxy (, ) ∂Bz (,tx ) ∂Btxz (, ) ∂Etxy (, ) ∂∂Etxyy(, ) Etx (, ) =− −=εμ00 = εμ00 ∂∂x t ∂∂x t ∂∂xt22 Speed of light in vacuum and medium; index of refraction n: 2 11 82 cm== −−12 2 2 7 2 ≈×(3 10 /s ) εμ00 (8.85×⋅× 10CNm / ) (4π 10 NA / ) 1 v2 ===≈ n cv// KK K v = cn εμ magn Relation between the amplitudes of the electric and magnetic fields in electromagnetic fields: E=cB. Radiation power: P=IA Intensity of radiation far away from the source: I = Pr/(4π 2 ) Density of energy: uExt= ε 2 (,); average density of energy uExtE==εε22(,) /2 0 urur 00 uruEB× rur EB Poynting vector S, intensity I: SPSAIS==⋅== μ002μ Radiation pressure: PIrad = α / c; for totally reflecting mirror α=2; for black body α=1. +++++++++++++++++++++++++++++++++++++++++++++++++++ Angle of reflection: θincident= θ reflected Snell’s law: nnincidentsinθ incident= refractedsinθ refracted nrefracted Angle of total internal reflection: sinθcritical = nincident r ur filter r r ur Polarizing by a linear filter along the direction nE: incident ⎯⎯⎯→⋅nnE()incident Malus’s law (consequence of the relation above): 2 II= max cos φ π π Polarizations: circular E== E ϕϕ ;;elliptical E≠= E ϕϕ yzyzmm22yzyz linear :.ϕϕyz= nrefracted Bruster’s angle: tgθ polar = nincident Huygens’s and Fermat’s principles. y ''s Images; lateral magnification: m = =− y s 11 1 Concave spherical mirror →), focal length: +=fR =||/2 ss' f 11 1 Convex spherical mirror →( : +=||||/2fR = ss'||− f nnnn− y'' ns Spherical refractive image abba+= m ==−a ss ' R y nsb 11 1s ' Thin lenses (converging lens, f>0; diverging lens, f<0): + ==−m ss' f s 11⎛⎞1 Lens maker’s equation: =−(1)n ⎜⎟ − f ⎝⎠RR12 11⎛⎞1 Double convex/concave lenses: =−(1)n ⎜⎟ + ||fRR | || | ⎝⎠12 ++++++++++++++++++++++++++++++++++++++++++++++ Integrals: 1 R 1 ∫ xN dx = x N +1 ∫ xN dx = R N +1 N +1 0 N + 1 1 1 1 ∞ 1 1 1 dx = dx = ∫ N N −1 ∫ N N −1 x −N + 1 x R x NR−1 1 b 1 ∫ dx= ln x ∫ dx= ln( b / a ) x a x a 11a xdy = ∫0 3/2 22 ()xy22+ x x + a t dtτ exp(−=−−ττ /00 ) τ (1 exp( / τ0 ) ∫0 Averaging 1 cos22 (ωωtkx−= ) sin ( tkx −= ) 2 cos(ωωtkx−⋅ ) sin( tkx − ) = 0 cos/63/2sin/61/2ππ== cosππ / 3== 1/ 2 sin / 3 3 / 2 .