Physics II EquationsCh. 21: Electric ChargesFe=k q1q2r2k =14 π ε0=8.99∗109N m2/C2ε0=8.85∗10−12C2/ N m2Ch. 22-23: Electric FieldsE=Fq0E=kqr2^rE=∑iEiEd=2 kqdz3τ =− pE sin θτ =p ×EU =−p ⦁E=− pE cos θE=k∫dqr2^rE=2 πkσ =σ2 ε0 (infinite plate)φ=E ⦁A (uniform field)φ=∮E ⦁dA∮E ⦁dA=qenε0E=λ2 π ε0 (infinite line of charge)E=σ2 ε0 (infinite sheet)E=ρr2 ε0 (uniform spherical shell)Ch. 24: Electric PotentialV =UqW =−q ∆U (by system)W =q ∆ V (on system)∆ V =−∫ifE ⦁ ds∆ K =−q ∆ V1 eV =1.602∗10−19J∆ V =−EdV =kqr (point charge)V =k∑iqiriV =kp cosθr2 (dipole)V =k∫dqrVdisk=2 πkσ (√a2+ y2− y)Es=− dVdsE=−∇ VCh. 25: Capacitanceq=CVC=ε0Ad(¿ plates )C=L2 k ln(ba)(cylinder)qseries=constantVs=∑Vi1Cs=∑1Ciqp=∑qiVp=constantCp=∑CiWext=q22 CU =q22 C=12C V2u=12ε0E2ED=E0κVD=V0κqD=q(1−1κ)CD= κ C0C=ε0Ad +t(1κ−1)∮κE⦁ dA=qenε0Ch. 26: Current and Resistanceiavg=∆ q∆ ti=dqdtJ=nevdV =iRJ=σE=1ρEσ =1ρR=ρLAP=iV =i2R=V2RCh. 27: CircuitsE=dWdq∑i=0( juntion rule)∑V =0(loop rule)Rs=∑i=1nRi1Rp=∑i=1n1Riq=C E(1−e−tRC)i=(ER)e−t / RCV =E(1−e−tRC)q=C E e−t / RC(discharging)Ch. 28: Magnetic FieldsF=qvB sin θF=qv ×BVH=iBnqtr=mvqB(cyclotron)T =2 πmqBf =qB2 πmF=iLB sin θF=iL ×Bτcoil=NiABsin θ=μB sinθτcoil=μ ×Bμ=NiA^nCh. 29: Magnetic Field due to CurrentB=μ0i2 πRμ0=4 π∗10−7Tm/ AdB=μ04 πids sinθr2dB=μ04 πids×^rr2Barc=μ0iφ4 πR∮B ⦁ ds=μ0ienB=μ0ir2 π R2(r <R)B=μ02 πμz3(dipole)B=μ0∋(solenoid )Ch. 30: Induction and InductanceΦ=B⦁A=BA cos θ=∫B ⦁ dAE=−NdΦdtE0=NBAωP=B2L2v2RL=NΦi=μ0n2Aln=(¿turnslength)∗(length)E=−Ldidti=ER(1−e−tτL)i=ERe−t / τL(decaying current)U =12L i2F=q(E+v +B)(Lorentz force)∮E ⦁ ds=−AdBdtLs=∑LiCh. 31: Electromagnetic Oscillations and AC Currentq=Q cos(ωt+φ)ω=1√LC=1RCi=I sin(ωt +φ)I=ωtU =L i22=q22 Cq(t)= Q e− Rt/ 2 Lcos(ω't+φ)ω'=√ω2−(R2 L)2VR=IRRVC=ICXCXC=1ωCVL=ILXLXL= ωLI=EmZZX=√R2+( XL−XC)2tan φ=XL− XCRIrms=I√2Vrms=V√2Erms=E√2Pavg=I2rmsRCh. 32: Maxwell’s Equations∮E ⦁ dA=qenε0Gauss’ Law for electricity∮B ⦁ dA=0Gauss’ Law for magnetism∮E ⦁ ds=−d ΦBdtFaraday’s Law∮B ⦁ ds=μ0ien+μ0ε0d ΦEdtAmpere-Maxwell Lawid=ε0d ΦEdtCh. 33: E-M Wavesc=3.00∗108m/sS=1μ0E ×BS=1μ0EB=E2c μ0I=1c μ0Erms2Erms=Em√2∆ p=∆ Uc∆ p=2 ∆ Ucif all light reflectedPr=FA=Ictot . absorptionPr=2 Ictot .reflectionI=I0cos2θn=cvn1sin θ1=n2sin θ2θi=θrsin θc=n2n1,n1>n2Ch. 16: Waves Iy(x)= ymsin kxk =2 πλv=fλ=ωk=λTy(x, t)= ymsin(kx −ωt +φ)∂ y (x , t)∂ t=−ω ymcos(kx−ωt +φ)∂2y ( x , t)∂ t2=−ω2ymsin(kx−ωtω+φ)sin α +sin β=2[cos12(α−β)][sin12(α+ β)]y(x, t)=2 ym[cos12φ]¿Wave interferencey(x, t)= 2 ym(cos ωt )¿Standing wavex=nλ2(nodes)x=(n+12)λ2λ=2 Ln(resonant standing wave)L=nλ2(resonant standing wave)Ch. 17: Waves II (Sound)L=nλ2Open pipef =nv2 LOpen pipeL=nλ41-open endf =nv2 L1-open endv=√Bρs(x , t)=smcos(kx−ωt)∆ P(x ,t)=Pmsin(kx−ωt )f'=fv ± vDv ± vS∆ L=mλFully constructive interferenceφ=2 mπFully constructive interference∆ L=(m+12)λFully destructive interferenceφ=(2m+1)πFully destructive interferenceω=12(ω1+ω2)ωbeat=ω1−ω2I=12ρv ω2sm2β=(10 dB)logII0Ch. 34: Imagesf =12r1p+1i=1fm=h'h=−ipn1p+n2i=n2−n1rCh. 35: Interferenced sin θ=mλConstructive Interference (maxima)d sin θ=(m+12)λDestructive Interference (minima)I=4 I0cos2(φ2)φ=2 πdλsin θφ= 2 mπ (maxima)φ=(2m+1)π (minima)λn=λn∆ φ=2 πλn∆ L+∆ φ02 L=(m+12)λn2Constructive Interference (Thin Film)2 L=mλn2Destructive Interference (Thin Film)Ch. 36: Diffractiona sin θ=mλI=Im(sin αα)2α=πaλsin θsin θ=1.22λd(Raleigh Criterion)∆ θhw=λNd cos θd sin θ=mλ2 d sinθ=mλX-Ray DiffractionCh. 37: Relativity∆ t0=2 Dc∆ t =∆ t0√1−v2c2β=vcγ=1√1−β2∆ t=γ ∆ t0L=L0γx'=γ (x−vt)y'= yz'=zt'=γ(t−vxc2)Lorentz Transformationsu=u'+v1+v u'c2f =f0√1−β1+βv=|∆ λ|λ0cp=γmvE0=m c21 u=1.66∗10−27kgE=E0+K=mc2+KE=γm c2Q=Mic2−Mfc2=−∆ M c2K= E−m c2=γmc2−mc2= m c2(γ −1)( pc)2=K2+2 Kmc2E2=(