Additional Formula Sheet for Final Exam eading and thoroughly familiarizing yourself with this formula sheet is an important part of, but it is not a substitute for, proper exam preparation. The latter requires, among other things, that you have re-worked all assigned homework problem sets (PS) and the in-class quizzes, studied the posted PS solutions, and worked and studied the assigned conceptual practice (CP) problems, as well as (optionally) some practice test (PT) problems, as posted on the LON-CAPA homework and on the PHYS1112 examples and homework web pages. You should consult the syllabus, and in particular review the Class Schedule on the last syllabus page (posted on the PYS1112 course web site), to find out which topics you should cover in preparing for this exam. Induction (1) Definition of Magnetic Flux: For a flat surface S with area vector A normal to the surface and finite surface area A A, subjected to a uniform magnetic field B, with angle θ ( A, B) enclosed between A and B, the magnetic flux through S is defined as: Φ m AB cos(θ). For surfaces S which are not flat and/or magnetic fields B which are not uniform across S, Φ m must be calculated by: (1) breaking up S into smaller, (approximately) flat pieces, with B being (appproximately) uniform across each such surface piece; and by then: (2) adding up the flux contributions from all these surface pieces. (2) Faraday s Law: Induced voltage (EMF) E in a tightly wound conducting coil of N identical turns, with each turn enclosing the same time-dependent magnetic flux Φ m (t): E = N Φ m Sign Convention. A positive EMF, E > 0, drives an induced current around the coil (or would do so if the open ends of the coil were connected into a closed circuit) in right-hand (H) direction around the A-vector that was used to define the flux Φ m (t). A negative EMF, E < 0, drives an induced current around the coil against H direction around the A-vector. (The H direction around A is the direction of the H 4 fingers when the H thumb points in A-direction.) The sign convention is equivalent to Lenz s ule: the induced current produces a magnetic field which opposes the change of the inducing magnetic field. (3) Motional EMF: Induced EMF E and electric field E in a conducting straight rod or wire of length l moving at speed v perpendicular to the rod/wire, in a uniform magnetic field B with field component B perpendicular to the rod/wire: E El = B vl. 1
Current I flowing through, magnetic breaking force F acting on, and mechanical power input P into (=heating power output from), a straight rod/wire with motionally induced EMF, connected into closed circuit of resistance : I = E/ = B vl, F = Il B = B2 vl 2, P = I 2 = F v = B2 v 2 l 2. (4) Generator-Induced EMF: The induced EMF of a coil of N turns, all of same of area A, rotating in uniform B-field at constant angular speed ω is E(t) = NABω sin(ωt) where θ(t) = ωt is the angle between B and the rotating area vector A(t), with A(t) normal to the planar surface enclosed by the turns of the coil. (5) Self-Induction: A time-dependent current I(t) in any wire or coil produces a selfinduced EMF E which opposes the change of the current and is proprtional to the rate of change of the current, E = L I/ I/, where the inductance L E I/ depends only on the shape and size of the wire or coil and is independent of E or I/. Inductance of a Long, Thin Solenoid (LTS). For a LTS of N turns over length l, with cross-sectional area A: L = µ o N 2 A l (6) Magnetic Field Energy: The energy U B required to build up a current I in an inductance L, stored as magnetic field energy, mostly inside the inductance coil, is given by U B = 1 2 LI2 Magnetic Field Energy Density. Energy per volume, u B, stored in a magnetic field is given in terms of the field strength B u B = 1 2µ o B 2 (7) Transformer: The primary (P ) and secondary (S) coil time-dependent currents (I P, I S ), induced voltages (V P, V S ), and number of turns (N P, N S ), are related by: V S V P = N S N P, I S I P = N P N S 2
(1) EM Field Energy Density: Electromagnetic (EM) Waves u = u E + u B = ɛ o 2 E2 + 1 2µ o B 2 (2) EM Wave Amplitudes, Intensity, Power and Speed: E- and B-field amplitudes, E max and B max, and time-averaged (TA) wave intensity I av, TA EM wave power P av transported through area A, and TA EM wave energy density u av are related by I av 1 U A 1 A P av = cu av = c ɛ o 2 E2 max = c 1 Bmax 2, B max = 1 2µ o c E max. Here, U is the energy transported through the area A, of a surface oriented perpendicular to EM wave propagation direction, during time. The speed of light c, ɛ o and µ o are related by c = 1 ɛo µ o (3) EM Wave Momentum and adiation Pressure: The momentum p carried by an EM wave packet of energy U is p = 1 c U The radiation pressure Π and average pressure force F av exerted by an EM wave of TA intensity I av, on an object exposed to cross-sectional area A of the EM wave beam, are related by Π 1 A F av 1 p A = α I av c Here, α = 1 for a 100% absorbing and α = 2 for a 100% reflecting object. Also, p is the EM wave momentum transferred through area A onto the object during time internal. (4) EM Wave Polarization: For an unpolarized EM wave beam of TA intensity I o incident upon a polarization filter, the TA intensity I transmitted through the filter is I = 1 2 I o Malus Law. For a polarized EM wave beam of TA intensity I o and electric field vector amplitude E o incident upon a polarization filter of transmission axis T, the TA intensity I and electric field vector amplitude E transmitted through the filter obey I = I o cos 2 (θ), E = E o cos(θ). Here, θ ( E o, T ) = ( E o, E) is the angle between the incident E o and T with E T ; and E o E o and E E. 3
Mechanics Memories: Velocity, Acceleration, Force, Energy, Power (1) Velocity (2) Acceleration v = r a = v r if constant; else v = lim 0 v if constant; else a = lim 0 (3) Constant-Acceleration Linear Motion: for r r f r i and v v f v i r = 1 2 ( v i + v f ) t ; r = v i t + 1 2 a t2 ; v = a t. (4) Constant-Speed Circular Motion: for motion at constant speed v v around a circular trajectory of radius r. The velocity vector v is always tangential to trajectory and perpendicular to acceleration vector a: v a. The acceleration vector a always points towards the center of the circular trajectory. Period T and frequency f of revolution, angular velocity ω, and orbital speed v: T = 1 f = 2πr v = 2π ω ω = 2πf = 2π T = v r v = ωr = 2πfr = 2πr T Circular centripetal acceleration: a = v2 r = ω2 r Orbital angle φ and arc of circumference s covered during time interval : φ = ω = v r (5) Newton s 2nd Law: = s r m a = F s = v = ω r = φ r (6) Kinetic Knergy (KE), Work, Work-KE-Theorem: K=kinetic energy of object of mass m moving at speed v; W =work done by force F on an object moving/moved with displacement r, with r pointing at an angle θ from F and 0 o θ 180 o ; K = K f K i = change of kinetic energy due to work done by total force F : K = 1 2 m v2, W = F r cos θ, K = W. (7) Energy Conservation Law for K K f K i and U U f U i : K i + U i = K f + U f or K + U = 0 4
(8) Mechanical Power: P =rate of work done by force F on an object moving at speed v, with v pointing at an angle θ from F and 0 o θ 180 o : P = F v cos θ. Algebra and Trigonometry az 2 + bz + c = 0 z = b ± b 2 4ac 2a sin θ = opp hyp, adj cos θ = hyp, opp tan θ = adj = sin θ cos θ For very small angles θ (with θ 90 o ): sin 2 θ + cos 2 θ = 1 sin θ = tan θ = θ (in radians) Numerical Data Acceleration of gravity (on Earth): g = 9.81m/s 2 Speed of light in vacuum: c = 3.00 10 8 m/s Coulomb s constant: k = 8.99 10 9 Nm 2 /C 2 Biot-Savart s constant: k m µo 4π = 1 10 7 Tm/A (exact) Permittivity of vacuum: ɛ o 1/(4πk) = 8.85 10 12 C 2 /Nm 2 Permeability of Vacuum: µ o 4πk m = 4π 10 7 Tm/A (exact) Elementary charge: e = 1.60 10 19 C Electron mass: Proton mass: m e = 9.11 10 31 kg m p = 1.67 10 27 kg Other numerical inputs will be provided with each problem statement. SI numerical prefixes: y = yocto =10 24, z = zepto =10 21, a = atto =10 18, f = femto =10 15, p = pico =10 12, n = nano =10 9, µ= micro =10 6, m = milli =10 3, c = centi =10 2, d = deci =10 1, da = deca =10 +1, h = hecto =10 +2, k = kilo =10 +3, M = Mega =10 +6, G = Giga =10 +9, T = Tera =10 +12, P = Peta =10 +15, E = Exa =10 +18, Z = Zetta =10 +21, Y = Yotta =10 +24. 5