Neet physics formula sheet, Exams of Physics

Download Neet physics formula sheet and more Exams Physics in PDF only on Docsity! 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 ε0 8.85× 10−12 F/m Coulomb constant 1 4πε0 9× 109 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| = √ a2 x + a2 y + a2 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+ 1 2at 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 − 1 2gt 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~p dt , ~F = m~a Newton’s third law: ~FAB = −~FBA Frictional force: fstatic, max = µsN, fkinetic = µkN Banking angle: v2 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 = 1 2mv 2 = p2 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 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 = h 3 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 C r r 2 6. Solid Hemisphere: yc = 3r 8 C r 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: 1 2m1v1 2+1 2m2v2 2 = 1 2m1v ′ 1 2 +1 2m2v ′ 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 − ω0 2 = 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 +md2 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~L dt , 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 = 1 2mvcm 2 + 1 2Icmω 2, ~L = Icm~ω + ~rcm ×m~vcm 1.7: Gravitation Gravitational force: F = Gm1m2 r2 m1 m2F F r Potential energy: U = −GMm r Gravitational acceleration: g = GM R2 Variation of g with depth: ginside ≈ g ( 1− h R ) Variation of g with height: goutside ≈ g ( 1− 2h R ) Effect of non-spherical earth shape on g: gat pole > gat equator (∵ Re −Rp ≈ 21 km) Effect of earth rotation on apparent weight: 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− x v ), I = I0 Spherical Wave: E = aE0 r sinω(t− r v ), I = I0 r2 Young’s double slit experiment Path difference: ∆x = dy D S1 P S2 d y D θ Phase difference: δ = 2π λ ∆x Interference Conditions: for integer n, δ = { 2nπ, constructive; (2n+ 1)π, destructive, ∆x = { nλ, constructive;( n+ 1 2 ) λ, destructive Intensity: I = I1 + I2 + 2 √ I1I2 cos δ, Imax = (√ I1 + √ I2 )2 , Imin = (√ I1 − √ I2 )2 I1 = I2 : I = 4I0 cos2 δ 2 , Imax = 4I0, Imin = 0 Fringe width: w = λD d Optical path: ∆x′ = µ∆x Interference of waves transmitted through thin film: ∆x = 2µd = { nλ, constructive;( n+ 1 2 ) λ, destructive. Diffraction from a single slit: θb y y D For Minima: nλ = b sin θ ≈ b(y/D) Resolution: sin θ = 1.22λ b Law of Malus: I = I0 cos2 θ I0 I θ 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: 1 v + 1 u = 1 f 3. Magnification: m = − v u 3.2: Refraction of Light Refractive index: µ = speed of light in vacuum speed of light in medium = c v Snell’s Law: sin i sin r = µ2 µ1 µ1 µ2 incident refracted reflected i r Apparent depth: µ = real depth apparent 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+δm 2 sin A 2 , 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: 1 f = (µ− 1) [ 1 R1 − 1 R2 ] Lens formula: 1 v − 1 u = 1 f , m = v u f u v Power of the lens: P = 1 f , 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 = v u 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−µr µy−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′ 4 Heat and Thermodynamics 4.1: Heat and Temperature Temp. scales: F = 32 + 9 5C, K = C + 273.16 Ideal gas equation: pV = nRT , n : number of moles van der Waals equation: ( p+ a V 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: F A = 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 = 1 3ρv 2 rms Equipartition of energy: K = 1 2kT for each degree of freedom. Thus, K = f 2kT for molecule having f de- grees of freedoms. Internal energy of n moles of an ideal gas is U = f 2nRT . 4.3: Specific Heat Specific heat: s = Q m∆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 = f 2RT , 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 = Q2 W = Q2 Q1−Q2 Entropy: ∆S = ∆Q T , Sf − Si = ∫ f i ∆Q T Const. T : ∆S = Q T , 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 = x KA Rseries = R1 +R2 = 1 A ( x1 K1 + x2 K2 ) x1 A x2 K1 K2 1 Rparallel = 1 R1 + 1 R2 = 1 x (K1A1 +K2A2) K1 K2 x A1 A2 Kirchhoff’s Law: emissive power absorptive 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: dT dt = −bA(T − T0) 5.7: Electromagnetic Induction Magnetic flux: φ = ∮ ~B ·d~S Faraday’s law: e = −dφ dt 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 = −Ldi dt Self inductance of a solenoid: L = µ0n 2(πr2l) Growth of current in LR circuit: i = e R [ 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 = 1 2Li 2 Energy density of B field: u = U V = B2 2µ0 Mutual inductance: φ = Mi, e = −M di dt 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 = 1 T ∫ 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 = 1 ωCR LR circuit: i L R e0 sinωt˜ R ωL Z φ Z = √ R2 + ω2L2, tanφ = ωL R 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: N1 N2 = e1 e2 , e1i1 = e2i2 i1 N1 i2 N2e1 ˜ e2 ˜ Speed of the EM waves in vacuum: c = 1/ √ µ0ε0 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 ) − φ e 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 8ε02h2n2 , En = −13.6Z2 n2 eV Radius of the nth Bohr’s orbit: rn = ε0h 2n2 πmZe2 , rn = n2a0 Z , a0 = 0.529 Å Quantization of the angular momentum: l = nh 2π 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: dN dt = −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 = Ic Ie , β = Ic Ib , β = α 1−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