Equations Sheet for Exam - The Physical World | PHY 182, Study notes of Physics

Download Equations Sheet for Exam - The Physical World | PHY 182 and more Study notes Physics in PDF only on Docsity! Physics 182 Equation Sheet for Exams • Good Stuff from 181: Kinematics: ~v = d~rdt ; ~a = d~v dt ; (~a = const : x = x0+v0∆t+ 1 2a∆t 2; v = v0+a∆t; v2 = v20+2a∆x) Dynamics: ~F = m~a Conservation of Energy: ∑ Wnc = ∆K + ∑ ∆U ; K = 12mv 2; Ugrav = mgy; Uspring = 12kx 2 Gravity: ~Fg = (Gm1m2r2 , attractive) Ug = −Gm1m2 r • Chapter 16: Moles : n = M(in g)Mmol = N NA Ideal Gas : PV = nRT = NkBT kB = RNA Temperatures: TF = 95TC + 32 ◦ TK = TC + 273 • Chapter 17: Work: W (on the system) = − ∫ Vf Vi P dV = −(area under PV curve) (general) = −P∆V (isobaric) = −nRT ln ( Vf Vi ) (isothermal) = − [ PfVf − PiVi (1− γ) ] (adiabatic) First Law: Solids/Liquids: ∆Eth = Q = Mc∆T Qf = ±MLf Qv = ±MLv First Law: Gases: ∆Eth = Q + W Specific Heats: (constant volume) Q = nCV ∆T (constant pressure) Q = nCP ∆T (any process) ∆Eth = nCV ∆T CP = CV + R Adiabatic Process: Q = 0 PV γ = constant TV γ−1 = constant Heat Transfer: Conduction: Q∆t = k A L ∆T Radiation: Qemit ∆t = eσAT 4 Qnet ∆t = eσA(T 4−T 4e ) • Chapter 18: Mean Free Path: λ = 1 4 √ 2π(N/V )r2 r ≈ .5× 10−10 m(monatomic), r ≈ 1× 10−10 m(diatomic) Kinetic Theory: P = 13 ( N V )mv 2 rms = 2 3 ( N V )²avg ²avg = 1 2mv 2 rms = 3 2kBT vrms = √ 3kBT m Ideal Gases: Eth = nCV T Monatomic: CV = 32R CP = 5 2R γ = CP CV = 53 Diatomic: CV = 52R CP = 7 2R γ = CP CV = 75 Solids: Eth = nCT C = 3R • Chapter 19: Work done by the system: Ws = −W (on the system) = + ∫ PdV Heat Engine: QH = QC + Wout η = WoutQH = 1− QC QH ηCarnot = 1− TCTH Refrigerator: Win + QC = QH K = QCWin = QC QH−QC KCarnot = TC TH−TC • Chapter 26: Coulomb Law: |Fq1q2 | = K |q1||q2|r2 = 14π²0 |q1||q2| r2 Charge in a Field: ~F = q ~E Field of a Point Charge q: ~E = K qr2 r̂ = 1 4π²0 q r2 r̂ • Chapter 27: Dipole Moment: ~p = (qs, from − q to + q) ~E(on axis ‖ ~p) ≈ 2K~pr3 ~E(on axis ⊥ ~p) ≈ −K~pr3 Continuous Charge Distribution Q: d ~E = K dqr2 r̂ ~E = ∫ Q d ~E Charge Densities: 1D : λ = Q/L 2D : η = Q/A 3D : ρ = Q/V Special Field Results: Infinite Wire: ~E = ( 2K|λ|r ; away for λ > 0, toward for λ < 0) 1 Infinite Plane: ~E = ( |η|2²0 ; away for η > 0, toward for η < 0) Parallel Plate Capacitor: ~E = ( η²0 , positive to negative) Ring of Charge: Ez = 14π²0 zQ (z2+R2)3/2 Disk of Charge: Ez = η2²0 [ 1− z√ z2+R2 ] • Chapter 28: Flux through Surface S: Φe = ∫ S ~E · d ~A Special Cases: Planar surface & uniform ~E: Φe = ~E · ~A Planar surface ⊥ uniform ~E: Φe = EA Gauss’ Law: For closed surface S: Φe = ∮ ~E · d ~A = Qin²0 Some Geometry: Cylinder (radius r, length L): A = 2πrL + 2πr2; V = πr2L Sphere (radius r): A = 4πr2; V = 43πr 3 • Chapter 29: Electric Potential Energy: Uniform field: U = qEs point charges: U = kq1q2r = 1 4π²0 q1q2 r Electric Potential: V = Uq ∆V = − ∫ ~E · d~s Special Cases: Uniform ~E: ∆V = − ~E · ~s Point Charge: V = kqr = 14π²0 q r Continuous Distribution: dV = kdqr • Chapter 30: Field and Potential: ∆V = VB−VA = − ∫ B A ~E ·d~s Ex = −∂V/∂x Ey = −∂V/∂y Ez = −∂V/∂z Capacitors: C = Q∆V Parallel Plate: C = ²0A d Energy: Uc = Q2 2C = 1 2C(∆V ) 2 Energy Density: uE = 12²0E 2 Combinations: Parallel: C = C1 + C2 Series: 1C = 1 C1 + 1C2 Dielectric: C = κC0 • Chapter 31: Electron Current: i = nAvd vd = eτme E Current: ~I = (dQ/dt, direction of ~E) I = ei = JA J = nevd ~J = σ ~E = ~E/ρ σ = 1ρ = ne2τ me Conservation of Current: Current is the same at any point along the wire. Resistance: R = ρ LA Ohm’s Law: I = ∆V R • Chapter 32: Kirchoff’s Laws Junction: ∑ Iin = ∑ Iout Closed Loop: ∑ i(∆V )i = 0 Conventions: EMF: ∆V = ±E ( from − → + from + → − ) Resistor: ∆V = ∓IR ( with I against I ) Power: Pbat = IE PR = I2R = (∆V ) 2 R = I∆V Combinations: Series: R = R1 + R2 Parallel: 1R = 1 R1 + 1R2 RC Circuit: Discharging: Q(t) = Q0e−t/τ I(t) = I0e−t/τ Time Constant: τ = RC Charging: Q(t) = Qmax(1− e−t/τ ) I(t) = ERe−t/τ 2