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PHYSICS 2000 : INTRODUCTION TO PHYSICS II FORMULA SHEET, Spring 2014 Coulomb’s Law: |F~ | = k|q1 q2 | r2 , k= 1 4π0 ~ = Electric field and force: E = 9.00 × 109 N · m2 /C2 ~ = k q2 r̂ Electric field due to a point charge: E a group of point charges: r Proton charge: e = 1.60 × 10−19 C ~ = R dE ~ = k R dq2 r̂ a continuous distribution of charge: E r ~ = k Pi E ~ F q0 qi r̂ ri2 i Electric dipole moment p~: |~p| = |q|d , two charges q and −q separated by a distance d. ~ the torque on an electric dipole is: ~ , In uniform E, ~τ = p~ × E ~ . and the potential energy of a dipole is: U = −~p · E 2kp θ Field due to the dipole, on its axis: Ez = z 3 , Potential due to dipole: V = kp rcos 2 Electric flux through a ~ plane surface in uniform E: Gauss’ Law: ΦE = ~ ·A ~ ΦE = E⊥ A = E H ~ · dA ~ = Qencl /0 , E ~ curved surface in any E: R ~ · dA ~ E net electric flux ΦE through any closed surface equals the net charge enclosed Qencl divided by 0 . 1 (0 = 4πk = 8.854 × 10−12 C2 /N · m2 ) ~ = k|Qencl |/r2 . |E| ~ = 2k|λencl |/r . |E| ~ = |σleft − σright |/20 . |E| For any spherically symmetric charge distribution, For any cylindrically symmetric charge distribution, For any plane-symmetric charge distribution, Change in potential energy: ∆U = Ub − Ua = −Wa→b = − Potential difference: ~ in a uniform E: ∆V = ∆U /q = − ~ · ~` , ∆V = −E Rb a ~ · d~` E Rb a R ~ · d~` F~ · d~` = −q ab E units: 1 volt = 1 V = 1 J/C ~` = displacement from a to b. V = k qr . Setting V = 0 at ∞, potential due to a point charge q a distance r from the charge: Potential energy of a pair of point charges q1 , q2 : ΦE = U12 = k qr112q2 , r12 = the distance of separation. Electric potential due to a continuous charge distribution: V = k ∂V ∂V ∂V Electric field from electric potential: Ex = − ∂x , Ey = − ∂y , Ez = − ∂z . R dq r . ~ = 0 everywhere inside. Conductor in electrostatic equilibrium: 1. E 2. excess charge resides on surface. ~ ~ 3. E just outside is perpendicular to the surface, |E| = σ/0 , where σ is the surface charge density. 4. surface is equipotential, & the potential is the same everywhere throughout the conductor. The capacitance of any capacitor: parallel plate capacitor: isolated charged sphere: C = |Q|/|Vab | , C = 0 A/d ; C = 4π0 R ; units: 1 farad = 1 F = 1 C/V. A= plate area, d= plate separation. R= radius. Equivalent capacitance Ceq capacitors in parallel: Ceq = C1 + C2 + C3 + . . . Potential energy stored in capacitor: U = Q2 2C capacitors in series: 1 Ceq = 1 C1 + 1 C2 + 1 C3 + ... ~ uE := U/Vol. = 0 E 2 /2 . = 12 QV = 12 CV 2 , energy density of E: C0 = capacitance without dielectric inserted between conductors, C= capacitance with dielectric inserted between conductors, C = KC0 , where K= the dielectric constant. E.g., parallel-plate capacitor: C = K0 A/d = A/d , := K0 is the permittivity of the dielectric; energy density in dielectric: u := K0 E 2 /2 = E 2 /2 ; H ~ · dA ~ = Qencl−free /0 , Gauss’ Law: K E where Qencl−free is the free charge (not bound charge) enclosed by the Gaussian surface. R ~ . , units: 1 A = 1 C/s . J~ current density, I = J~ · dA Electric current: I = dQ dt Conductor, cross-sectional area A, charge carriers of charge q, density n, drift velocity ~vd , the current is I = nqvd A, so that the current density is J~ = nq~vd . ~ , where ρ is the resistivity of the material. Ohm’s Law: I = V /R and J~ = E/ρ The resistance R of a conductor is R = ρ`/A . Unit of R: 1 ohm = 1 Ω. Temperature dependence: ρ(T ) = ρ0 [1 + α(T − T0 )] , where α is the temperature coefficient of resistivity of the material. The power, or rate at which energy is absorbed by an electrical device, is P = IVab . Emf E = dW dq Equivalent resistance Req resistors in series: Req = R1 + R2 + R3 + . . . resistors in parallel: R1eq = R11 + 1 R2 + 1 R3 + ... P Junction Rule: I = 0 , sum of currents entering a node = sum of currents leaving that node. P Loop Rule: V = 0 , sum of potential differences across each element around a closed-loop circuit = 0. RC Circuits: capacitor charging up: Q(t) = Qf [1 − e−t/RC ], I(t) = I0 e−t/RC ; capacitor discharging: Q(t) = Q0 e−t/RC , I(t) = (Q0 /RC)e−t/RC . ~ Magnetic force on a particle of charge q moving with velocity ~v in a magnetic field B: ~ 1 tesla = 1 T = 1 N/(A·m) . Units of magnetic field B: ~ + q~v × B ~ . When both magnetic and electric fields are present, F~ = q E ~ . F~ = q~v × B ~ F~ = I ~`0 × B, ~ Force on a conductor carrying current I in a uniform magnetic field B: ~`0 starts where the current enters the conductor, and ends where it exits. R ~ . In a nonuniform magnetic field, F~ = I d~` × B ~ where A ~ is a vector whose The magnetic moment µ ~ of a loop of N turns carrying a current I is µ ~ = N I A, magnitude equals the area of the loop, and whose direction is perpendicular to the loop. ~ experiences a torque ~τ = µ ~ , A current loop (or any magnetic moment) in a uniform magnetic field B ~ ×B ~ and its potential energy is U = −~µ · B. ~ = µ0 q~v×r̂ Magnetic field of a moving charge: B . 4π r2 ~ Biot-Savart law: Magnetic field dB due to a current element Id~` carrying a steady current I is: ~ = µ0 Id~`×r̂ , where µ0 = 4π × 10−7 T·m/A and r is the distance from the element to the field. dB 4π r2 ~ = R dB ~ = µ0 R Id~`×r̂ The total magnetic field due to the entire conductor is B . 4π r2 µ0 I Magnetic field a distance a from a long, straight wire carrying a current I is B = 2πa . The force per unit length between two parallel wires a distance a apart, carrying currents I1 and I2 , is I1 I2 F = µ02πa ; the force is attractive (repulsive) for currents in the same (opposite) directions. ` H ~ · d~` = µ0 Iencl . That is, the line integral of B ~ · d~` around any Ampere’s Law states B closed path (or Amperian loop) equals µ0 times Iencl , the total steady current encircled by that path. From Ampere’s law, the field inside an ideal solenoid is B = µ0 N` I = µ0 nI . ~ = µ0 |Iencl |/(2πr) . For any cylindrically symmetric current distribution, |B| ~ = µ0 |(I/`)left − (I/`)right |/2 . For any plane-symmetric charge distribution, |B| ~ varies with time), The generalized form of Ampere’s Law (generalized to situations in which E or the Ampere-Maxwell Law: H ~ · d~` = µ0 (IC + ID )encl = µ0 Iencl + µ0 0 dΦE , B dt where ΦE is the electric field flux through any surface bounding the closed path, ID := 0 dΦdtE is the displacement current, and Ic = I is the conduction current. R ~ · dA ~ . Magnetic flux through a surface: ΦB = B H ~ · dA ~ = 0, Gauss’ Law of magnetism: B the net magnetic flux through any closed surface is zero (there are no magnetic monopoles). H ~ · d~` = − dΦB . Faraday’s Law of Induction: emf E = E dt That is, the emf induced around a closed path (or loop) has magnitude equal to the time rate of change of the magnetic flux through the loop. For N turns, the induced emf is N times as big. Lenz’s Law: the emf induced around a loop has the direction that creates, or would create, a magnetic field that opposes the change in ΦB that produced them. ~ Motional emf: when a straight conducting bar of length vector ~` moves through a magnetic field B R ~ · ~` . In general, E = (~v × B) ~ · d~`. with a velocity ~v , the induced emf is E = (~v × B) induced emf: E2 = −M dIdt1 mutual inductance: M = N2IΦ1B2 emf of an inductor: emf = −L dI , self-inductance L = N ΦIB (SI unit: 1 henry=1 H); dt ideal solenoid: L = µ0 N 2 A/` . Energy stored in an inductor: U = LI 2 /2 , energy density of a magnetic field: uB = U/Vol. = B 2 /2µ0 . CALCULUS R n R −1 d xn+1 m m−1 x = m x , x dx = (n = 6 1) , x dx = ln |x| dx n+1 R d d cos x = − sin x , sin x = cos x , cos x dx = sin x , dx R x dx d x x e = e = e dx dx ) df (x) d chain rule: dx g f (x) = dg(f df dx R sin x dx = − cos x