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Formula Sheet for LSU Physics 2113, Third Exam, Fall ’14 • Constants, definitions: m g = 9.8 2 s m3 G = 6.67 × 10−11 kg · s2 MSun = 1.99 × 1030 kg C2 o = 8.85 × 10−12 Nm2 8 c = 3.00 × 10 m/s REarth = 6.37 × 106 m MEarth = 5.98 × 1024 kg RM oon = 1.74 × 106 m Earth-Sun distance = 1.50×1011 m MM oon = 7.36 × 1022 kg 1 Nm2 = 8.99 ×109 2 k= C 4πo mp = 1.67 × 10−27 kg Earth-Moon distance = 3.82×108 m dipole moment: p ~ = q d~ me = 9.11 × 10−31 kg Area of a circle: A = πr 2 Area of a cylinder: A = 2πr` Area of a sphere: A = 4πr 2 Volume of a cylinder: V = πr 2 ` e = 1.60 × 10−19 C 1 eV = e(1V) = 1.60×10−19 J Q Q Q charge densities: λ = , σ = , ρ = L A V Volume of a sphere: V = 43 πr 3 • Units: Joule = J = N·m • Kinematics (constant acceleration): v = vo + at x − xo = 21 (vo + v)t • Circular motion: mv 2 Fc = mac = , r T = 2πr v , x − xo = vo t + 21 at2 v 2 = vo2 + 2a(x − xo ) v = ωr • General (work, def. of potential energy, kinetic energy): ~net = m~ K = 12 mv 2 F a Emech = K + U W = −∆U (by field) Wext = ∆U = −W (if objects are initially and finally at rest) • Gravity: ~| = G Newton’s law: |F m1 m2 r2 Gravitational Field: ~ g = −G Law of periods: T2 4π 2 = GM M r2 ! rˆ = − r dVg dr 3 Potential Energy of a System (more than 2 masses): I GM Gravitational acceleration (planet of mass M ): ag = r2 GM Gravitational potential: Vg = − r m1 m2 Potential Energy: U = −G r12 m1 m2 m1 m3 m2 m3 U =− G +G +G + ... r12 r13 r23 ~ = −4πGMins ~ g · dS Gauss’ law for gravity: S • Electrostatics: ~| = k Coulomb’s law: |F | q1 || q2 | ~ = qE ~ Force on a charge in an electric field: F r2 |q| ~ =k Electric field of a point charge: |E| r2 2k~ p ~ = Electric field of a dipole on axis, far away from dipole: E z3 2kλ ~ = Electric field of an infinite line charge: |E| r ~ Torque on a dipole in an electric field: ~ τ =p ~×E ~ ~ Potential energy of a dipole in E field: U = −~ p·E • Electric flux: Φ = R ~ · dA ~ E • Gauss’ law: o I ~ · dA ~ = qenc E • Electric field of an infinite non-conducting plane with a charge density σ: E = σ 2o • Electric field of infinite conducting plane or close to the surface of a conductor: E = σ o • Electric potential, potential energy, and work: Vf − Vi = − Z f ~ · d~ E s i ~ ~ = −5V, E Ex = − ~ · ∆~ In a uniform field: ∆V = −E s = −Ed cos θ ∂V ∂x , Ey = − ∂V , Ez = − ∂V ∂y ∂z n n X X qi q Potential of n point charges: V = Vi = k Potential of a point charge q: V = k r r i=1 i=1 i Electric potential energy: ∆U = q∆V ∆U = −Wfield q1 q2 Potential energy of two point charges: U12 = Wext = q2 V1 = q1 V2 = k r12 • Capacitance: definition: q = CV Capacitor with a dielectric: C = κCair Potential Energy in Cap: U = Capacitors in parallel: Ceq • Current: i = dq dt Z = q2 2C P = Ci = 1 2 Parallel plate: C = ε◦ qV = 1 2 CV 2 ~ Const. curr. density: J = J~ · dA, • Definition of resistance: R = V i L A i A , Charge carrier0 s drift speed: ~ vd = J~ ne ~ |E| |J~| Temperature dependence: ρ − ρ◦ = ρ◦ α(T − T◦ ) Power dissipated in a resistor: P = i2 R = • Power in an electrical device: P = iV V2 R dW dq • Resistors in series: Req = P Ri d ~ 2 Energy density of electric field: u = κεo |E| 2 X 1 1 Capacitors in series: = Ceq Ci Definition of resistivity: ρ = • Resistance in a conducting wire: R = ρ • Definition of emf : E = 1 A Resistors in parallel: 1 Req = X 1 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 • RC circuit: Charging: q(t) = CE(1 − e− τc ), Time constant τC = RC, Discharging: q(t) = qo e− τc • Magnetic Fields: ~ = q~ ~ Magnetic force on a charge q: F v×B i Hall voltage: V = vd Bd = B d = width ⊥ to field and i, nle ~ = qE ~ + q~ ~ Lorentz force: F v×B l = thickness k to field and ⊥ to i Circular motion in a magnetic field: qv⊥ B = 2 mv⊥ with period: T = r ~ = iL ~ ×B ~ Magnetic force on a length of wire: F ~ Magnetic Dipole: µ ~ = N iA ~ Torque: ~ τ =µ ~ ×B (µ0 = 4π × 10−7 • Generating Magnetic Fields: 4π ~ Potential energy: U = −~ µ·B T·m ) A r3 Magnetic field of a long straight wire: B = µ0 2i 4π r Magnetic field of a circular arc: B = Force between parallel current carrying wires: Fab = I qB µ0 id~ s×~ r ~ = Biot-Savart Law: dB Ampere’s law: 2πm ~ · d~ B s = µ0 ienc Magnetic field of a toroid: B = µ0 ia ib 2πd µ0 i 4π r φ L Magnetic field of a solenoid: B = µ0 in µ0 iN 2πr ~ = , Magnetic field of a dipole on axis, far away: B µ0 µ ~ 2π z 3 • Induction: Z ~ · dA ~ B Magnetic Flux: Φ = Faraday’s law: E = − dΦ dt I Induced Electric Field: (= −N dΦ dt ~ · d~ E s=− Definition of Self-Inductance: L = dΦ dt NΦ Inductance of a solenoid: L = µ0 n2 Al i EMF (Voltage) across an inductor: E = −L RL Circuit: Rise of current: i = Magnetic Energy: UB = 1 2 Li2 E R Motional emf: E = BLv for a coil with N turns) tR di dt (1 − e− L ), Time constant: τL = L R tR , Decay of current: i = i0 e− L Magnetic energy density: uB = B2 2µ0