Formula Sheet for LSU Physics 2113, Third Exam, Fall '14 • Constants, definitions: m g = 9:8 R = 6:37 × 106 m M = 5:98 × 1024 kg s2 Earth Earth m3 G = 6:67 × 10−11 R = 1:74 × 106 m Earth-Sun distance = 1.50×1011 m kg · s2 Moon 30 22 8 MSun = 1:99 × 10 kg MMoon = 7:36 × 10 kg Earth-Moon distance = 3.82×10 m 2 2 −12 C 1 9 Nm −19 o = 8:85 × 10 2 k = = 8.99 ×10 2 e = 1:60 × 10 C Nm 4πo C 8 −19 c = 3:00 × 10 m/s mp = 1:67 × 10−27 kg 1 eV = e(1V) = 1.60×10 J −31 Q Q Q dipole moment: ~p = qd~ me = 9:11 × 10 kg charge densities: λ = ; σ = ; ρ = L A V 2 2 4 3 Area of a circle: A = πr Area of a sphere: A = 4πr Volume of a sphere: V = 3 πr Area of a cylinder: A = 2πr` Volume of a cylinder: V = πr2` • Units: Joule = J = N·m • Kinematics (constant acceleration): 1 1 2 2 2 v = vo + at x − xo = 2 (vo + v)t x − xo = vot + 2 at v = vo + 2a(x − xo) • Circular motion: mv2 2πr Fc = mac = , T = , v = !r r v • General (work, def. of potential energy, kinetic energy): 1 2 ~ K = 2 mv Fnet = m~a Emech = K + U W = −∆U (by field) Wext = ∆U = −W (if objects are initially and finally at rest) • Gravity: m1m2 GM Newton's law: jF~j = G Gravitational acceleration (planet of mass M): ag = r2 r2 M dVg GM Gravitational Field: ~g = −G r^ = − Gravitational potential: Vg = − r2 dr r 2 ! 4π m1m2 Law of periods: T 2 = r3 Potential Energy: U = −G GM r12 m1m2 m1m3 m2m3 Potential Energy of a System (more than 2 masses): U = − G + G + G + ::: r r r I 12 13 23 Gauss' law for gravity: ~g · dS~ = −4πGMins S • Electrostatics: j q1 jj q2 j Coulomb's law: jF~j = k Force on a charge in an electric field: F~ = qE~ r2 j q j Electric field of a point charge: jE~j = k r2 2k~p Electric field of a dipole on axis, far away from dipole: E~ = z3 2kλ Electric field of an infinite line charge: jE~j = r Torque on a dipole in an electric field: ~τ = ~p × E~ Potential energy of a dipole in E~ field: U = −~p · E~ I R • Electric flux: Φ = E~ · dA~ • Gauss' law: o E~ · dA~ = qenc σ • Electric field of an infinite non-conducting plane with a charge density σ: E = 2o σ • Electric field of infinite conducting plane or close to the surface of a conductor: E = o • Electric potential, potential energy, and work: Z f Vf − Vi = − E~ · d~s In a uniform field: ∆V = −E~ · ∆~s = −Ed cos θ i @V @V @V E~ = −5~ V; Ex = − ;Ey = − ;Ez = − @x @y @z n n q X X qi Potential of a point charge q: V = k Potential of n point charges: V = Vi = k r i=1 i=1 ri Electric potential energy: ∆U = q∆V ∆U = −Wfield q1q2 Potential energy of two point charges: U12 = Wext = q2V1 = q1V2 = k r12 • Capacitance: definition: q = CV A Capacitor with a dielectric: C = κCair Parallel plate: C = "◦ d 2 q 1 1 2 1 2 Potential Energy in Cap: U = = qV = CV Energy density of electric field: u = κεojE~j 2C 2 2 2 P 1 X 1 Capacitors in parallel: Ceq = Ci Capacitors in series: = Ceq Ci Z ~ dq i 0 J • Current: i = = J~·dA~, Const. curr. density: J = , Charge carrier s drift speed: ~vd = dt A ne V jE~j • Definition of resistance: R = Definition of resistivity: ρ = i jJ~j L • Resistance in a conducting wire: R = ρ Temperature dependence: ρ − ρ◦ = ρ◦α(T − T◦) A V 2 • Power in an electrical device: P = iV Power dissipated in a resistor: P = i2R = R dW • Definition of emf : E = dq P 1 X 1 • Resistors in series: Req = Ri Resistors in parallel: = Req Ri • Loop rule in DC circuits: the sum of changes in potential across any closed loop of a circuit must be zero. • Junction rule in DC circuits: the sum of currents entering any junction must be equal to the sum of currents leaving that junction. − t − t τc τc • RC circuit: Charging: q(t) = CE(1 − e ), Time constant τC = RC, Discharging: q(t) = qoe • Magnetic Fields: Magnetic force on a charge q: F~ = q~v × B~ Lorentz force: F~ = qE~ + q~v × B~ i Hall voltage: V = vdBd = B d = width ? to field and i, l = thickness k to field and ? to i nle 2 mv? 2πm Circular motion in a magnetic field: qv?B = with period: T = r qB Magnetic force on a length of wire: F~ = iL~ × B~ Magnetic Dipole: ~µ = NiA~ Torque: ~τ = ~µ × B~ Potential energy: U = −~µ · B~ T · m • Generating Magnetic Fields: (µ = 4π × 10−7 ) 0 A µ0 id~s × ~r Biot-Savart Law: dB~ = 4π r3 µ0 2i µ0 i Magnetic field of a long straight wire: B = Magnetic field of a circular arc: B = φ 4π r 4π r µ0iaib Force between parallel current carrying wires: Fab = L 2πd I Ampere's law: B~ · d~s = µ0ienc Magnetic field of a solenoid: B = µ0in µ0iN µ0 ~µ Magnetic field of a toroid: B = , Magnetic field of a dipole on axis, far away: B~ = 2πr 2π z3 • Induction: Z Magnetic Flux: Φ = B~ · dA~ dΦ dΦ Faraday's law: E = − (= −N for a coil with N turns) Motional emf: E = BLv dt dt I dΦ Induced Electric Field: E~ · d~s = − dt NΦ 2 Definition of Self-Inductance: L = Inductance of a solenoid: L = µ0n Al i di EMF (Voltage) across an inductor: E = −L dt E − tR L − tR RL Circuit: Rise of current: i = (1 − e L ), Time constant: τ = , Decay of current: i = i0e L R L R 2 1 2 B Magnetic Energy: UB = Li Magnetic energy density: uB = 2 2µ0.