PHY Tables & Formulas. You may refer to this handout on quizzes & exams. Do not add additional information. m

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1 PHY Tables & Formulas You may refer to this handout on quizzes & exams. Do not add additional information. m Things you should know from PHY 131 and other prerequisites. (If you don t, learn them now.) - Definitions of trig functions - Pythagorean theorem - Newton's three laws - Relationship between weight and mass - Formula for kinetic energy - Relationship between frequency and period - Definition of pressure - Ideal gas law Sec. 1: Gravitational Force: F Gm m r Field lines show direction of the force on a positive test charge. Superposition: Force: Vector sum of forces from all charges at other points. E : Vector sum of fields from all charges at other points. Sec. 2: Memorize: Definition of potential (and potential difference). B Potential difference: ΔV = V B - V A = E ds = E s if E constant A Potential energy of one charge: U = Σ U for every pair which includes that charge Total U of a group of charges: Σ U for every pair of charges. Potential at some point: V = Σ V from all charges at other points

2 - 2 - Charge per unit length: λ = Q/L Charge per unit area: σ = Q/A Charge per unit volume: ρ = Q/V Sec 4: Memorize: Definition of current, definition of capacitance, and Ohm s law. Between parallel plates in a vacuum: V E and d C = ε o A/d E Q 0 A Current density: Magnitude: J = I/A Direction: direction positive charge is moving If n = number of free charges per unit volume, q = charge of one of them, and v d = their drift velocity, J = nqv d Resistivity, ρ: E = ρj, ρ = RA/l Conductivity: σ = 1/ρ Series V tot = V 1 + V 2 + V 3 + I tot = I 1 = I 2 = I 3 = R eq = R 1 + R 2 + R Q tot = Q 1 = Q 2 = Q 3 = Parallel V tot = V 1 = V 2 = V 3 = I tot = I 1 + I 2 + I 3 + 1/R eq = 1/R 1 + 1/R Q tot = Q 1 + Q 2 + Q 3 + 1/C eq = 1/C 1 + 1/C 2 + C eq = C 1 + C 2 + C 3 + Sec 5: Electric Power: P = VI = I 2 R = V 2 /R (Recall, P = W/t = (energy delivered)/t) EMF (Electromotive "force", in volts): E = (energy form source)/q Loop rule: Σ potential changes around a closed loop = 0 (increase = pos, decrease = neg, where + side is higher V) Point rule: Total current into a point = total current out.

3 sec. 6: Unit vector = (No other formulas are new.) Sec. 7: Know how s direction is defined. Force on a moving charge: F q( v B) Charge circling in a B field: F centripetal = F magnetic (F c =mv 2 /r) Force on a wire: F I ( B) 's magnitude = wire's length, 's direction = direction of current Lorentz force equation: F qe q( v B) Sec. 8: (If field or angle vary from point to point,.) In a toroidal coil of B = μ o IN circumference 2πr: 2πr Sec. 9: Magnetic flux, Φ B : Replace E with B in Φ E equation, sec. 3

4 - 4 - Maxwell s Equations (If field or angle vary from point to point, replace with.): Sec. 10: Lenz's Law: Direction of induced current is such that coil's own B field opposes the change in flux. Permittivity: ε = κε o (ε replaces ε o in all equations when a material medium is present.) Magnetic permeability: μ replaces μ o in all equations when a material medium is present.) Sec. 11: LR circuits: Time constant: τ = L/R = time to get about 63% (1-1/e) of the way to the final value. Current decay: I = I 0 e -t/τ RC circuits: Time constant: τ = RC Discharging: q = Q 0 e -t/τ) Charging up: Find current from I = dq/dt, capacitor s voltage using C = q/v. (When discharging, Q o /C = V o.) Energy stored in a capacitor : U = ½ CV 2 = Q 2 /(2C) Energy per unit volume in an E field: u = ½ ε E 2 Energy stored in an inductor: U = ½ LI 2 Energy per unit volume in a B field: u = B 2 /(2μ)

5 - 5 - Find current from I = dq/dt, and voltage using C = q/v. (Q max /C = V max.) LC circuits total energy: U = ½ CV 2 + ½ LI 2 Resistor: V = IR (That is, V RMS = I RMS R and v max = i max R), v & i in phase. Capacitor: V = IX c, capacitive reactance = X c = 1/(ωC), ELI ICE Inductor: V = IX L, inductive reactance = X L = ωl Circuit as a whole: V = IZ, impedance = Z = Power: P AV = (I RMS ) 2 R = V RMS I RMS cos (cos = "power factor") Transformer: V 1 /V 2 = N 1 /N 2 and V 1 I 1 = V 2 I 2 Sec. 13: From Phy 131: Heat flow when temperature changes: Q = m c ΔT Heat flow when state changes: Q = m L f or Q = m L v First Law of Thermo.: E int = Q + W W = Work Done on System. E int = Increase in Internal Energy Q = Heat Added Molar Heat Capacity: Q = nc T For any Ideal Gas: Eint ncv T, C P - C v = R, Cv C P C P = constant pressure heat capacity, C V = constant volume heat capacity

6 - 6 - For a Monatomic Ideal Gas: 3 C V 2 R 5, C P R 2 Isobaric Process = No Pressure Change SEC. 14: Know 2 nd law of Thermodynamics in terms of entropy and in term of heat. Work Done In A Cycle = Area Enclosed On P-V Diagram. Q in = W + Q out Efficiency: Efficiency of Carnot Cycle: Change in Entropy for a Reversible Process:

7 Geometric Formulas: (r = radius, h = height) Circumference of a circle or sphere πr Area of circle πr 2 Area of a circular cylinder (excluding ends).... 2πrh Area of a sphere πr 2 Volume of a circular cylinder πr 2 h Volume of a sphere (4/3)πr 3 Quadratic Formula: If ax 2 + bx + c = 0 then x b b 2 2a 4ac Logarithms: def.: if x = b y then log b (x) = y common log: log = log 10 natural log: ln = log e where e = Properties (for log of any base): log (xy) = log x + log y log (x a ) = a log x log (x/y) = log x - log y log b (b x ) = x Short Table of Integrals: x n dx = x n+1 + c if n =/ -1 sin(ax)dx = - 1 cos ax + c n+1 a dx/x = ln x + c e ax dx = 1 e ax + c a cos(ax)dx = 1 sin(ax) + c a sec 2 (ax)dx = 1 tan(ax) + c a sin 2 (ax)dx = x - sin(2ax) + c cos 2 (ax)dx = x + sin(2ax) +c 2 4a 2 4a sin(ax)cos(ax)dx = sin 2 (ax) + c 2a

8 Partial Derivatives: Differentiate with respect to one variable, treating the others as constants. example: z = x + x 2 y 3 z = 1 + (2x)y 3 z = 0 + x 2 (3y 2 ) x y Vector Multiplication: Determinants: Material Dielectric Dielectric Constant, k Strength (V/m) Vacuum Air x 10 6 Bakelite x 10 6 Fused quartz x 10 6 Pyrex glass x 10 6 Polystyrene x 10 6 Teflon x 10 6 Neoprene rubber x 10 6 Nylon x 10 6 Paper x 10 6 Strontium titanate x 10 6 Water 80 - Silicone oil x 10 6

9 Molar Heat Capacities of gasses: Monatomic: Diatomic: Polyatomic: C P C V C P C V C P C V (J/mol K) (J/mol K) (J/mol K) (J/mol K) (J/mol K) (J/mol K) He H CO Ar N SO Ne O H 2 O Kr CO CH Cl Elementary charge: e = x C Permittivity of free space: ε o = x c 2 /N m 2 1 = x 10 9 N m 2 /c Gravitational constant: G = x N m 2 /kg 2 Speed of light in a vacuum: c = x 10 8 m/s Electron mass: m e = x kg Proton mass: m p = x kg Permeability of free space: μ o = 4π x 10-7 N/A 2 Universal Gas Constant: R = J/mole K Some physical constants: SI prefixes: Power: Prefix: Abbreviation: Power: Prefix: Abbreviation: yocto y 10 1 deka da zepto z 10 2 hecto h atto a 10 3 kilo k femto f 10 6 mega M pico p 10 9 giga G 10-9 nano n tera T 10-6 micro μ peta P 10-3 milli m exa E 10-2 centi c zetta Z 10-1 deci d yotta Y

10 Fundamental Units: Standard SI Unit and Abbreviation: Conversion Factors LENGTH meter = m 1 m = 3.28 ft, 1 mile = 1609 m = 5280 ft 1 Angstrom = m, 1 inch = 2.54 cm TIME second = s 1 hour = 3600 s, 1 day = 86,400 s 1 year = 3.16 x 10 7 s MASS kilogram = kg 1kg=.0685 slug, weighs 2.21 lb in standard gravity 1 unified mass unit = 1 u = 1.66 x kg CURRENT ampere = C/s = A TEMPERATURE kelvin = k T (in kelvins) = T (in Celsius) Derived Units VOLUME m 3 1 Liter = 10-3 m 3 = cm 3 SPEED m/s 1 mi/hr = m/s = 1.47 ft/sec FORCE newton = N 1 N = pound ENERGY joule = J 1 calorie = J 1 J = ft lb & WORK 1 electronvolt = x J, 1 BTU = 252 cal POWER watt = W 1 horsepower = 746 W = 550 ft lb/sec ANGLE radian = rad 1 revolution = 360 = 2π rad FREQUENCY hertz = Hz 1 Hz = 60 rev/min = 1 cycle/sec IMPULSE & MOMENTUM kg m/s = N S PRESSURE pascal = N/m 2 1 atmosphere = x 10 5 Pa = 14.7 lb/in 2 CHARGE coulomb = C ELECTRIC FIELD N/C = V/m POTENTIAL CAPACITANCE volt = V farad = F RESISTANCE REACTANCE& IMPEDANCE ohm = Ω ELECTRIC FLUX N m 2 /C = V m MAGNETIC FLUX weber = Wb MAGNETIC FIELD Wb/m 2 = tesla = T INDUCTANCE henry = H 1 T = 10 4 gauss