formula.pdftest 1 solutions 2011.pdfProblem 1 flux pp 1-4.pdfFlux 1.pdfFlux 2.pdfFlux 3.pdfFlux 4.pdfProblem 2 el field pp 1-4.pdfEl field pp 1 2.pdfpage 1 El field.pdfpage 2 El field.pdfProblem 2a.pdfProblem 2b.pdfProblem 3 el potent pp 1-2.pdfEl pot 1.pdfEl pot 2.pdfProblem 4.pdftest 2 solutions 2011.pdf1.pdf2.pdf3.pdf4.pdf5.pdf5a.pdftest 3 solutions 2011.pdf3-1.PDF3-2.PDF3-3.PDF3-4.PDF3-5.PDF3-6.PDF3-7.PDF208 Formulae Sheet Coulomb’s law: . Electric field created by a charge q: 204 rqEπε=r 22F1041rqqπε=r Permittivity of free space: 2292290/109/10941CmNcoulombmeterNewton ⋅⋅=⋅×=πε Gauss’s law (electric flux through a closed surface): . Surface area of a sphere of radius R is 24 RSπ=A jump of the electric field over a charged surface: 0||||Eσδε=r Electric potential of a point charge q: () ( )rqVrV04πε=∞−r. Unit: 1volt=J/C ε0 = 8.85×10-12 C/(Vm) Definition of the electric potential difference: Conservation of energy for a charge Q: constrQVK=+)(. Energy of an electron in electric potential=1volt (electron volt): 1eV=1.6×10-19 J (J=1Joule). Capacitance: C=Q/V; Parallel-plate capacitor: ε =K ε0 Spherical capacitor: 11 1 14inner outerCRRπε⎛⎞=−⎜⎟⎝⎠; 4abbaRRCRRπε=− Unit: ε0 = 8.85×10-12 F/m Capacitors in parallel: Capacitors in series: Capacitor as energy storage 2222QCVUC== u=energy density 2()()2Erurε=rr +++++++++++++++++++++++++++++++++++++++++++++++++ Definition of current dtdQtQtIt=ΔΔ=→Δ 0lim)(. Unit: 1A=1ampere= C/s. 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 Resistor with a constant cross section: sec 'Length LRcross tion s area Sρρ==−. Resistivityρis measured in [Ωm]. ∫⋅eddEε=SenclosQS0rr∫=⋅SenclosedQSdE0εrr0(when an axes is directed from left to right !)EEσεrhs lhs−=2121[() ()]rrEdr V r V r=− −∫rrrrrrdAVQCε= =Δ11 1 /F farad coulomb volt≡=12 6110 110pFFFμ−−==F3...1111321+++=CCCCtot...21+++= CCCtotCResistors in series 321RRRRtot++= +… Resistors in parallel ...1111321+++=RRRRtot Similarity between resistance and capacitance: 1/ 1/RCρε⇔⇔ Power output (energy loss rate): . Unit: [J/s] 22/PIVRI VR== =Discharging capacitor: ( ) exp( / )initialqt Q t RC=−()/qtIdqdtRC==−; negative I implies that the charge flows out from the plate, i.e., it is discharging Charging capacitor ( ) [1 exp( / )]finalqt Q t RC=−−() exp( / )finalQIttRC=−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. ++++++++++++++++++++++++++++++++++++++++++++++ Force acting on a charge q moving in the magnetic field Force acting on an element dl of a current-carrying conductor: Cyclotron frequency: Dipoles. Electric dipole moment of a pair charges separated by : ; dur Magnetic dipole moment of a small area surrounded by a current I: Torque [Nm]: pEτ=×rurru; Bτμ=×rurur. Energy of a dipole in a field: UpE=−⋅urur; UBμ=−⋅urur Magnetic field created by a moving charge q (Biot-Savart law): Magnetic field created by an element dl carrying current I: Units for magnetic field Permeability Magnetic field created by a straight wire carrying current I : Steady-state version of Ampere’s law (current enclosed by a path): Magnetic field created by a solenoid:0BnIμ=, 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 through the loop): Ampere’s law (including “displacement” current created by varying in time electric fields): 03()4qv rBrμπ×=rrrBvqFrrr×=BlIdFrr×=22qBfmωππ==pqd=ur ur±q()IdSμ=ur ur03()4Idl rdBrμπ×=rrrmANsmCNteslaT⋅=⋅=/1//1)(177220410 / 410 / 410 /TmA Ns C NAμπ π π−−=× ⋅ =× ⋅ =×72−0I||2Brμ=π∫contour0 enclosedBdl Iμ=ur r∫Φ−=⋅dtdrdEBrr00()EcdBdr IdtμεΦ=+urr∫Maxwell’s equations: two Gauss’s laws + Faraday’s and Ampere’s laws Mutual Inductance: 222 222BBd11Emf N N M IdtΦ=− Φ = BMIΦ=21 12MM= 111 1112BBd2Emf N N M IdtΦ=− Φ = Mutual Inductance: 012mutual overlap overlapMnnlSμ= Units for flux (weber) and EMF: 22[]1 1 /1/11111/Bflux T m N m s C J s C V sTm Wb V WbsΦ=⋅=⋅⋅=⋅=⋅⋅= = Units of the mutual inductance (henry): 11 21/11/111/henry H Wb A V s A s J A== =⋅ =Ω⋅= BddEmf N Ldt dtIΦ=− =−Inductance (self-inductance): Another units for permeability: 70410 /Hmμπ−=×Inductance of a toroidal solenoid: 202BNNLAIrμπΦ== ×rea Current growth in an R-L circuit: (1 exp( / ))EmfIRt LR=−− Decay of current in an R-L circuit: ( 0)exp( / )IIt Rt L==− Magnetic field energy: 22() ( / )()22LI t dQ dtUt L==Density of magnetic field energy 202Buμ=B Oscillations in a L-C circuit:2210dqqdt LC+= , ; 21/ LCω=2221()()222() sin( )qt QLIt constCCIt I tωϕ+===+ ++++++++++++++++++++++++++++++++++++++++++++++ Waves (frequency, wave vector, speed): 2π 2πλω =k=v==ω/k ω =vkT λ T Wave propagating along x: /(;) cos( )right lefty x t A kx t phaseωϕϕ=+=m Wave equation: 22222(, ) (, )ytx ytxvxt∂∂=∂∂ Set of wave equations in electromagnetism: Speed of light in vacuum and medium; index of refraction n: 2 8212 2 2 7 20011(3 10 / )(8.85 10 / ) (4 10 / )cmCNm NAεμ π−−== ≈××⋅×s 00(, )(, )yzEtxBtxxtεμ∂∂−=∂∂220022(, ) (,yyEtx Etxxtεμ∂∂=∂∂(, )(, )yzEtxBtxxt∂∂=−∂∂)21//magnv n cv KK K v cnεμ=== ≈ =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: 2/(4 )IPrπ= Density of energy: ; average density of energy20(,)uExtε=2200(,) /2uExtEεε== Poynting vector S, intensity I: Radiation pressure: /radPIcα= ; for totally reflecting mirror α=2; for black body α=1. 002EB ESPSAISBμμ×==⋅=u=rurururur +++++++++++++++++++++++++++++++++++++++++++++++++++ Angle of reflection: incident