Physics Important Formulae, Study notes of Physics

Download Physics Important Formulae and more Study notes Physics in PDF only on Docsity! Formulae Sheet for Physics www.concepts-of-physics.com | pg. 1 Physics formulas from Mechanics, Waves, Optics, Heat and Thermodynamics, Electricity and Magnetism and Modern Physics. Also includes the value of Physical Constants. Helps in quick revision for CBSE, NEET, JEE Mains, and Advanced. 0.1: Physical Constants Speed of light c 3× 108 m/s Planck constant h 6.63× 10−34 J s hc 1242 eV-nm Gravitation constant G 6.67×10−11 m3 kg−1 s−2 Boltzmann constant k 1.38× 10−23 J/K Molar gas constant R 8.314 J/(mol K) Avogadro’s number NA 6.023× 1023 mol−1 Charge of electron e 1.602× 10−19 C Permeability of vac- uum µ0 4π × 10−7 N/A2 Permitivity of vacuum 0 8.85× 10−12 F/m Coulomb constant 14π0 9× 10 9 N m2/C2 Faraday constant F 96485 C/mol Mass of electron me 9.1× 10−31 kg Mass of proton mp 1.6726× 10−27 kg Mass of neutron mn 1.6749× 10−27 kg Atomic mass unit u 1.66× 10−27 kg Atomic mass unit u 931.49 MeV/c2 Stefan-Boltzmann constant σ 5.67×10−8 W/(m2 K4) Rydberg constant R∞ 1.097× 107 m−1 Bohr magneton µB 9.27× 10−24 J/T Bohr radius a0 0.529× 10−10 m Standard atmosphere atm 1.01325× 105 Pa Wien displacement constant b 2.9× 10−3 m K 1 MECHANICS 1.1: Vectors Notation: ~a = ax ı̂+ ay ̂+ az k̂ Magnitude: a = |~a| = √ a2x + a 2 y + a 2 z Dot product: ~a ·~b = axbx + ayby + azbz = ab cos θ Cross product: ~a ~b~a×~b θ ı̂ ̂k̂ ~a×~b = (aybz−azby )̂ı+(azbx−axbz)̂+(axby−aybx)k̂ |~a×~b| = ab sin θ 1.2: Kinematics Average and Instantaneous Vel. and Accel.: ~vav = ∆~r/∆t, ~vinst = d~r/dt ~aav = ∆~v/∆t ~ainst = d~v/dt Motion in a straight line with constant a: v = u+ at, s = ut+ 12at 2, v2 − u2 = 2as Relative Velocity: ~vA/B = ~vA − ~vB Projectile Motion: x y O u si n θ u cos θ u θ R H x = ut cos θ, y = ut sin θ − 12gt 2 y = x tan θ − g 2u2 cos2 θ x2 T = 2u sin θ g , R = u2 sin 2θ g , H = u2 sin2 θ 2g 1.3: Newton’s Laws and Friction Linear momentum: ~p = m~v Newton’s first law: inertial frame. Newton’s second law: ~F = d~pdt , ~F = m~a Newton’s third law: ~FAB = −~FBA Frictional force: fstatic, max = µsN, fkinetic = µkN Banking angle: v 2 rg = tan θ, v2 rg = µ+tan θ 1−µ tan θ Centripetal force: Fc = mv2 r , ac = v2 r Pseudo force: ~Fpseudo = −m~a0, Fcentrifugal = −mv 2 r Minimum speed to complete vertical circle: vmin, bottom = √ 5gl, vmin, top = √ gl Conical pendulum: T = 2π √ l cos θ g mg T l θ θ 1.4: Work, Power and Energy Work: W = ~F · ~S = FS cos θ, W = ∫ ~F ·d~S Kinetic energy: K = 12mv 2 = p 2 2m Potential energy: F = −∂U/∂x for conservative forces. Ugravitational = mgh, Uspring = 1 2kx 2 Work done by conservative forces is path indepen- dent and depends only on initial and final points:∮ ~Fconservative ·d~r = 0. Work-energy theorem: W = ∆K Get Formulas www.concepts-of-physics.com c© 2020 by Jitender Singh Ver. 2020 1 Get Our Book Formulae Sheet for Physics www.concepts-of-physics.com | pg. 2 Mechanical energy: E = U +K. Conserved if forces are conservative in nature. Power Pav = ∆W ∆t , Pinst = ~F ·~v 1.5: Centre of Mass and Collision Centre of mass: xcm = ∑ ximi∑ mi , xcm = ∫ xdm∫ dm CM of few useful configurations: 1. m1, m2 separated by r: m1 m2 C r m2r m1+m2 m1r m1+m2 2. Triangle (CM ≡ Centroid) yc = h3 C h 3 h 3. Semicircular ring: yc = 2r π C 2r πr 4. Semicircular disc: yc = 4r 3π C 4r 3πr 5. Hemispherical shell: yc = r 2 Cr r 2 6. Solid Hemisphere: yc = 3r 8 Cr 3r 8 7. Cone: the height of CM from the base is h/4 for the solid cone and h/3 for the hollow cone. Motion of the CM: M = ∑ mi ~vcm = ∑ mi~vi M , ~pcm = M~vcm, ~acm = ~Fext M Impulse: ~J = ∫ ~F dt = ∆~p Collision: m1 m2 v1 v2 Before collision After collision m1 m2 v′1 v ′ 2 Momentum conservation: m1v1+m2v2 = m1v ′ 1+m2v ′ 2 Elastic Collision: 12m1v1 2+12m2v2 2 = 12m1v ′ 1 2 +12m2v ′ 2 2 Coefficient of restitution: e = −(v′1 − v′2) v1 − v2 = { 1, completely elastic 0, completely in-elastic If v2 = 0 and m1  m2 then v′1 = −v1. If v2 = 0 and m1  m2 then v′2 = 2v1. Elastic collision with m1 = m2 : v ′ 1 = v2 and v ′ 2 = v1. 1.6: Rigid Body Dynamics Angular velocity: ωav = ∆θ ∆t , ω = dθ dt , ~v = ~ω × ~r Angular Accel.: αav = ∆ω ∆t , α = dω dt , ~a = ~α× ~r Rotation about an axis with constant α: ω = ω0 + αt, θ = ωt+ 1 2αt 2, ω2 − ω02 = 2αθ Moment of Inertia: I = ∑ imiri 2, I = ∫ r2dm ring mr2 disk 1 2mr 2 shell 2 3mr 2 sphere 2 5mr 2 rod 1 12ml 2 hollow mr2 solid 1 2mr 2 rectangle m(a2+b2) 12 a b Theorem of Parallel Axes: I‖ = Icm +md 2 cm I‖ d Ic Theorem of Perp. Axes: Iz = Ix + Iy x yz Radius of Gyration: k = √ I/m Angular Momentum: ~L = ~r × ~p, ~L = I~ω Torque: ~τ = ~r × ~F , ~τ = d~Ldt , τ = Iα O x y P ~r ~F θ Conservation of ~L: ~τext = 0 =⇒ ~L = const. Equilibrium condition: ∑ ~F = ~0, ∑~τ = ~0 Kinetic Energy: Krot = 1 2Iω 2 Dynamics: ~τcm = Icm~α, ~Fext = m~acm, ~pcm = m~vcm K = 12mvcm 2 + 12Icmω 2, ~L = Icm~ω + ~rcm ×m~vcm 1.7: Gravitation Gravitational force: F = Gm1m2r2 m1 m2F F r Potential energy: U = −GMmr Gravitational acceleration: g = GMR2 Variation of g with depth: ginside ≈ g ( 1− hR ) Variation of g with height: goutside ≈ g ( 1− 2hR ) Effect of non-spherical earth shape on g: gat pole > gat equator (∵ Re −Rp ≈ 21 km) Effect of earth rotation on apparent weight: Get Formulas www.concepts-of-physics.com c© 2020 by Jitender Singh Ver. 2020 1 Get Our Book Formulae Sheet for Physics www.concepts-of-physics.com | pg. 5 5. 2nd overtone/5th harmonics: ν2 = 5ν0 = 5v 4L 6. Only odd harmonics are present. Open organ pipe: L A N A N A 1. Boundary condition: y = 0 at x = 0 Allowed freq.: L = nλ2 , ν = n v 4L , n = 1, 2, . . . 2. Fundamental/1st harmonics: ν0 = v 2L 3. 1st overtone/2nd harmonics: ν1 = 2ν0 = 2v 2L 4. 2nd overtone/3rd harmonics: ν2 = 3ν0 = 3v 2L 5. All harmonics are present. Resonance column: l 1 + d l 2 + d l1 + d = λ 2 , l2 + d = 3λ 4 , v = 2(l2 − l1)ν Beats: two waves of almost equal frequencies ω1 ≈ ω2 p1 = p0 sinω1(t− x/v), p2 = p0 sinω2(t− x/v) p = p1 + p2 = 2p0 cos ∆ω(t− x/v) sinω(t− x/v) ω = (ω1 + ω2)/2, ∆ω = ω1 − ω2 (beats freq.) Doppler Effect: ν = v + uo v − us ν0 where, v is the speed of sound in the medium, u0 is the speed of the observer w.r.t. the medium, consid- ered positive when it moves towards the source and negative when it moves away from the source, and us is the speed of the source w.r.t. the medium, consid- ered positive when it moves towards the observer and negative when it moves away from the observer. 2.4: Light Waves Plane Wave: E = E0 sinω(t− xv ), I = I0 Spherical Wave: E = aE0r sinω(t− r v ), I = I0 r2 Young’s double slit experiment Path difference: ∆x = dyD S1 P S2 d y D θ Phase difference: δ = 2πλ ∆x Interference Conditions: for integer n, δ = { 2nπ, constructive; (2n+ 1)π, destructive, ∆x = { nλ, constructive;( n+ 12 ) λ, destructive Intensity: I = I1 + I2 + 2 √ I1I2 cos δ, Imax = (√ I1 + √ I2 )2 , Imin = (√ I1 − √ I2 )2 I1 = I2 : I = 4I0 cos 2 δ 2 , Imax = 4I0, Imin = 0 Fringe width: w = λDd Optical path: ∆x′ = µ∆x Interference of waves transmitted through thin film: ∆x = 2µd = { nλ, constructive;( n+ 12 ) λ, destructive. Diffraction from a single slit: θb y y D For Minima: nλ = b sin θ ≈ b(y/D) Resolution: sin θ = 1.22λb Law of Malus: I = I0 cos 2 θ I0 I θ Visit www.concepts-of-physics.com to buy “IIT JEE Physics: Topic-wise Complete Solutions” and our other books. Written by IITians, Foreword by Dr. HC Verma, Appreciated by Students. Get Formulas www.concepts-of-physics.com c© 2020 by Jitender Singh Ver. 2020 1 Get Our Book Formulae Sheet for Physics www.concepts-of-physics.com | pg. 6 3 Optics 3.1: Reflection of Light Laws of reflection: normal incident reflectedi r (i) Incident ray, reflected ray, and normal lie in the same plane (ii) ∠i = ∠r Plane mirror: d d (i) the image and the object are equidistant from mir- ror (ii) virtual image of real object Spherical Mirror: O I u v f 1. Focal length f = R/2 2. Mirror equation: 1v + 1 u = 1 f 3. Magnification: m = − vu 3.2: Refraction of Light Refractive index: µ = speed of light in vacuumspeed of light in medium = c v Snell’s Law: sin isin r = µ2 µ1 µ1 µ2 incident refracted reflected i r Apparent depth: µ = real depthapparent depth = d d′ O Id d′ Critical angle: θc = sin −1 1 µ θc µ Deviation by a prism: µ δ i i′ A r r′ δ = i+ i′ −A, general result µ = sin A+δm2 sin A2 , i = i′ for minimum deviation δm = (µ− 1)A, for small A i δ δm i′ Refraction at spherical surface: P O Q µ1 µ2 u v µ2 v − µ1 u = µ2 − µ1 R , m = µ1v µ2u Lens maker’s formula: 1f = (µ− 1) [ 1 R1 − 1R2 ] Lens formula: 1v − 1 u = 1 f , m = v u f u v Power of the lens: P = 1f , P in diopter if f in metre. Two thin lenses separated by distance d: 1 F = 1 f1 + 1 f2 − d f1f2 f1 f2 d 3.3: Optical Instruments Simple microscope: m = D/f in normal adjustment. Compound microscope: O ∞ Objective Eyepiece u v fe D 1. Magnification in normal adjustment: m = vu D fe 2. Resolving power: R = 1∆d = 2µ sin θ λ Astronomical telescope: fo fe 1. In normal adjustment: m = − fofe , L = fo + fe 2. Resolving power: R = 1∆θ = 1 1.22λ 3.4: Dispersion Cauchy’s equation: µ = µ0 + A λ2 , A > 0 Dispersion by prism with small A and i: 1. Mean deviation: δy = (µy − 1)A 2. Angular dispersion: θ = (µv − µr)A Dispersive power: ω = µv−µrµy−1 ≈ θ δy (if A and i small) Dispersion without deviation: µ µ′A A′ (µy − 1)A+ (µ′y − 1)A′ = 0 Deviation without dispersion: (µv − µr)A = (µ′v − µ′r)A′ Get Formulas www.concepts-of-physics.com c© 2020 by Jitender Singh Ver. 2020 1 Get Our Book Formulae Sheet for Physics www.concepts-of-physics.com | pg. 7 4 Heat and Thermodynamics 4.1: Heat and Temperature Temp. scales: F = 32 + 95C, K = C + 273.16 Ideal gas equation: pV = nRT , n : number of moles van der Waals equation: ( p+ aV 2 ) (V − b) = nRT Thermal expansion: L = L0(1 + α∆T ), A = A0(1 + β∆T ), V = V0(1 + γ∆T ), γ = 2β = 3α Thermal stress of a material: FA = Y ∆l l 4.2: Kinetic Theory of Gases General: M = mNA, k = R/NA Maxwell distribution of speed: v n vp v̄ vrms RMS speed: vrms = √ 3kT m = √ 3RT M Average speed: v̄ = √ 8kT πm = √ 8RT πM Most probable speed: vp = √ 2kT m Pressure: p = 13ρv 2 rms Equipartition of energy: K = 12kT for each degree of freedom. Thus, K = f2kT for molecule having f de- grees of freedoms. Internal energy of n moles of an ideal gas is U = f2nRT . 4.3: Specific Heat Specific heat: s = Qm∆T Latent heat: L = Q/m Specific heat at constant volume: Cv = ∆Q n∆T ∣∣∣ V Specific heat at constant pressure: Cp = ∆Q n∆T ∣∣∣ p Relation between Cp and Cv: Cp − Cv = R Ratio of specific heats: γ = Cp/Cv Relation between U and Cv: ∆U = nCv∆T Specific heat of gas mixture: Cv = n1Cv1 + n2Cv2 n1 + n2 , γ = n1Cp1 + n2Cp2 n1Cv1 + n2Cv2 Molar internal energy of an ideal gas: U = f2RT , f = 3 for monatomic and f = 5 for diatomic gas. 4.4: Theromodynamic Processes First law of thermodynamics: ∆Q = ∆U + ∆W Work done by the gas: ∆W = p∆V, W = ∫ V2 V1 pdV Wisothermal = nRT ln ( V2 V1 ) Wisobaric = p(V2 − V1) Wadiabatic = p1V1 − p2V2 γ − 1 Wisochoric = 0 Efficiency of the heat engine: T1 T2 Q1 Q2 W η = work done by the engine heat supplied to it = Q1 −Q2 Q1 ηcarnot = 1− Q2 Q1 = 1− T2 T1 Coeff. of performance of refrigerator: T1 T2 Q1 Q2 W COP = Q2W = Q2 Q1−Q2 Entropy: ∆S = ∆QT , Sf − Si = ∫ f i ∆Q T Const. T : ∆S = QT , Varying T : ∆S = ms ln Tf Ti Adiabatic process: ∆Q = 0, pV γ = constant 4.5: Heat Transfer Conduction: ∆Q∆t = −KA ∆T x Thermal resistance: R = xKA Rseries = R1 +R2 = 1 A ( x1 K1 + x2K2 ) x1 A x2 K1 K2 1 Rparallel = 1R1 + 1 R2 = 1x (K1A1 +K2A2) K1 K2 x A1 A2 Kirchhoff’s Law: emissive powerabsorptive power = Ebody abody = Eblackbody Wien’s displacement law: λmT = b λ Eλ λm Stefan-Boltzmann law: ∆Q∆t = σeAT 4 Newton’s law of cooling: dTdt = −bA(T − T0) Get Formulas www.concepts-of-physics.com c© 2020 by Jitender Singh Ver. 2020 1 Get Our Book Formulae Sheet for Physics www.concepts-of-physics.com | pg. 10 5.7: Electromagnetic Induction Magnetic flux: φ = ∮ ~B ·d~S Faraday’s law: e = −dφdt Lenz’s Law: Induced current create a B-field that op- poses the change in magnetic flux. Motional emf: e = Blv − + ~vl ⊗ ~B Self inductance: φ = Li, e = −Ldidt Self inductance of a solenoid: L = µ0n 2(πr2l) Growth of current in LR circuit: i = eR [ 1− e− t L/R ] e L R iS t i L R 0.63 eR Decay of current in LR circuit: i = i0e − t L/R L R iS t i i0 L R 0.37i0 Time constant of LR circuit: τ = L/R Energy stored in an inductor: U = 12Li 2 Energy density of B field: u = UV = B2 2µ0 Mutual inductance: φ = Mi, e = −M didt EMF induced in a rotating coil: e = NABω sinωt Alternating current: t i T i = i0 sin(ωt+ φ), T = 2π/ω Average current in AC: ī = 1T ∫ T 0 i dt = 0 RMS current: irms = [ 1 T ∫ T 0 i2 dt ]1/2 = i0√ 2 t i2 T Energy: E = irms 2RT Capacitive reactance: Xc = 1 ωC Inductive reactance: XL = ωL Imepedance: Z = e0/i0 RC circuit: i C R e0 sinωt˜ R 1 ωC Z φ Z = √ R2 + (1/ωC)2, tanφ = 1ωCR LR circuit: i L R e0 sinωt˜ R ωL Z φ Z = √ R2 + ω2L2, tanφ = ωLR LCR Circuit: i L C R e0 sinωt˜ R 1 ωC ωL Z 1 ωC − ωLφ Z = √ R2 + ( 1 ωC − ωL )2 , tanφ = 1 ωC−ωL R νresonance = 1 2π √ 1 LC Power factor: P = ermsirms cosφ Transformer: N1N2 = e1 e2 , e1i1 = e2i2 i1 N1 i2 N2e1 ˜ e2 ˜ Speed of the EM waves in vacuum: c = 1/ √ µ00 Visit www.concepts-of-physics.com to buy “IIT JEE Physics: Topic-wise Complete Solutions” and our other books. Written by IITians, Foreword by Dr. HC Verma, Appreciated by Students. Get Formulas www.concepts-of-physics.com c© 2020 by Jitender Singh Ver. 2020 1 Get Our Book Formulae Sheet for Physics www.concepts-of-physics.com | pg. 11 6 Modern Physics 6.1: Photo-electric effect Photon’s energy: E = hν = hc/λ Photon’s momentum: p = h/λ = E/c Max. KE of ejected photo-electron: Kmax = hν − φ Threshold freq. in photo-electric effect: ν0 = φ/h Stopping potential: Vo = hc e ( 1 λ ) − φe 1 λ V0 φ hc hc e −φe de Broglie wavelength: λ = h/p 6.2: The Atom Energy in nth Bohr’s orbit: En = − mZ2e4 802h2n2 , En = − 13.6Z2 n2 eV Radius of the nth Bohr’s orbit: rn = 0h 2n2 πmZe2 , rn = n2a0 Z , a0 = 0.529 Å Quantization of the angular momentum: l = nh2π Photon energy in state transition: E2 − E1 = hν E1 E2 hν Emission E1 E2 hν Absorption Wavelength of emitted radiation: for a transition from nth to mth state: 1 λ = RZ2 [ 1 n2 − 1 m2 ] X-ray spectrum: λmin = hc eV λ I λmin λα Kα Kβ Moseley’s law: √ ν = a(Z − b) X-ray diffraction: 2d sin θ = nλ Heisenberg uncertainity principle: ∆p∆x ≥ h/(2π), ∆E∆t ≥ h/(2π) 6.3: The Nucleus Nuclear radius: R = R0A 1/3, R0 ≈ 1.1× 10−15 m Decay rate: dNdt = −λN Population at time t: N = N0e −λt O t N0 N N0 2 t1/2 Half life: t1/2 = 0.693/λ Average life: tav = 1/λ Population after n half lives: N = N0/2 n. Mass defect: ∆m = [Zmp + (A− Z)mn]−M Binding energy: B = [Zmp + (A− Z)mn −M ] c2 Q-value: Q = Ui − Uf Energy released in nuclear reaction: ∆E = ∆mc2 where ∆m = mreactants −mproducts. 6.4: Vacuum tubes and Semiconductors Half Wave Rectifier: ˜ D R Output Full Wave Rectifier: ˜ Output Triode Valve: Filament Plate Grid Cathode Plate resistance of a triode: rp = ∆Vp ∆ip ∣∣∣ ∆Vg=0 Transconductance of a triode: gm = ∆ip ∆Vg ∣∣∣ ∆Vp=0 Amplification by a triode: µ = − ∆Vp∆Vg ∣∣∣ ∆ip=0 Relation between rp, µ, and gm: µ = rp × gm Current in a transistor: Ie = Ib + Ic Ic Ib Ie α and β parameters of a transistor: α = IcIe , β = Ic Ib , β = α1−α Transconductance: gm = ∆Ic ∆Vbe Logic Gates: AND OR NAND NOR XOR A B AB A+B AB A + B AB̄ + ĀB 0 0 0 0 1 1 0 0 1 0 1 1 0 1 1 0 0 1 1 0 1 1 1 1 1 0 0 0 Get Formulas www.concepts-of-physics.com c© 2020 by Jitender Singh Ver. 2020 1 Get Our Book