sign in sign up
<<
Home , Coulomb

208 Formulae Sheet

r 1 1qq 2 r q Coulomb’s law: F = 2 . created by a charge q: E = 2 4πε0 r 4πε 0r 1 of free space: ×= 109 9 ⋅meter 2 / coulomb2 ⋅⋅= /109 CmN 229 4πε 0

r r Q SdE =⋅ enclosed ∫ ε S 0 r r Q Gauss’s law (electric through a closed surface): ⋅ dE S = enclos ed . ∫ ε Surface area of a sphere of radius R is = 4πRS 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 of a point charge q: ()r VrV ( )=∞− . 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: + )( = constrQVK . Energy of an in electric potential=1volt (electron ): 1eV=1.6×10-19 J (J=1Joule).

Q A : C=Q/V; Parallel-plate : C = = ε ε =K ε0 ΔV d 11⎛⎞ 1 1 RabR Spherical capacitor: =−⎜⎟; C = 4πε CRR4πε ⎝⎠inner outer Rba− R Unit: 11F≡= 1 coulomb / volt −− 12 6 110110pFFF==μ F -12 ε0 = 8.85×10 F/m 1111 in parallel: C = + + CCC + ... Capacitors in series: +++= ... tot 21 3 CCCC tot 321 r QCV22 r E 2 ()r Capacitor as energy storage U == u=energy density ur()= ε 22C 2

+++++++++++++++++++++++++++++++++++++++++++++++++ ΔQ dQ Definition of current tI = lim)( = . Unit: 1A=1ampere= C/s. t→Δ 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]. 1111 Resistors in series tot += + RRRR 321 +… Resistors in parallel +++= ... tot RRRR 321 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 drops and rises along each of the closed loops is zero. ++++++++++++++++++++++++++++++++++++++++++++++

r r Force acting on a charge q moving in the F r ×= Bvq

r r Force acting on an element dl of a current-carrying conductor: ×= BlIdF

ω qB Cyclotron frequency: f == 22π πm ur ur ur Dipoles. 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 = ⋅ = /1//1)(1 ⋅mANsmCNteslaT

−−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 ’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): rdE r −=⋅ 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 () and EMF: B 1111/Tm⋅=2 WbV = Wbs Units of the mutual inductance (): 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 : 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 Eyzyz== E ϕϕmm;;elliptical Eyzyz≠= E ϕϕ 22 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 RR ⎝⎠12 11⎛⎞1 Double convex/concave lenses: =−(1)n ⎜⎟ + ||fRR | || | ⎝⎠12 ++++++++++++++++++++++++++++++++++++++++++++++ Integrals:

1 R 1 ∫ N dxx = x N +1 ∫ N dxx = R N +1 N +1 0 N + 1

11 1 ∞ 11 1 dx = dx = ∫ N N −1 ∫ N N −1 x +− 1 xN R x −1 RN

1 b 1 ∫ = ln xdx ∫ = abdx )/ln( 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