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Electric Forces and Fields
Coulomb’s Law: F = ke |q^1 rq 22 |
Superposition of Forces: F⃗net =
P
i
Fi Force from Electric Field: F⃗ = q⃗E
Electric Field at a Point: E⃗ = ⃗ Fq
Field of Infinite Line: E⃗ = (^2) πϵλ 0 r ˆr
Field of Infinite Plane: E⃗ = 2 σϵ 0 nˆ
Dipole Moment:p⃗ = q⃗d Torque on a Dipole:τ⃗ =p⃗ × E⃗
Gauss’s Law
Definition of Electric Flux: ΦE =
R
E · d⃗A Electric Flux for Uniform Electric Field: ΦE = EA cos θ Electric Flux through an Open Surface: ΦE =
R
S
E^ ·^ d⃗A Electric Flux through a Closed Surface: ΦE =
H
S
E^ ·^ d⃗A Gauss’s Law:
H
S
E^ ·^ d⃗A^ =^
Qenc ϵ 0 Gauss’s Law for Systems with Symmetry:
H
S
E^ ·d⃗A^ =^
Qenc ϵ 0 Electric Field Outside a Conductor: E = (^) ϵσ 0
Electric Potential and Potential Energy
Potential Energy of a Two-Charge System: U = keq r^1 q^2 Work Done to Assemble a System of Charges: W =
P
i<j
keqiqj rij Potential Difference: V = VB − VA = −
R B
A
E^ ·^ d⃗r Electric Potential: V = kreq Potential Difference Between Two Points: VAB = VB − VA = −
R B
A
E^ ·^ d⃗r Electric Potential of a Point Charge: V = kreq Electric Potential of a System of Point Charges: V =
P
i
keqi ri Electric Dipole Moment:p⃗ = q · d⃗ Electric Potential Due to a Dipole: V = (^4) πϵ^101 r (2 cos θ · p) Electric Potential of a Continuous Charge Distribution: V = (^4) πϵ^10
R (^) dq r Electric Field Components: E⃗ = −∇V Del Operator in Cartesian Coordinates: ∇ = ˆi (^) ∂x∂ + ˆj (^) ∂y∂ + ˆk (^) ∂z∂
Electric Field as Gradient of Potential: E⃗ = −∇V Del Operator in Cylindrical Coordinates: ∇ = ˆer ∂r∂ + ˆeϕ (^1) r∂ϕ∂ + ˆez ∂z∂ Del Operator in Spherical Coordinates: ∇ = ˆer ∂r∂ + ˆeθ (^1) r∂θ∂ + ˆeϕ (^) r sin^1 θ∂ϕ∂
Capacitance
Capacitance of a Parallel-Plate Capacitor: C = ϵ^0 dA Capacitance of a Vacuum Spherical Capacitor: C = 4 πϵ^0 R 1 − R R^12 Capacitance of a Vacuum Cylindrical Capacitor: C = 2 πϵ 0 L ln
(^) R 2 R 1
Capacitance of a Series Combination: (^) C^1 eq =
P
i
1 Ci Capacitance of a Parallel Combination: Ceq =
P
i Ci Energy Density: u = 12 ϵ 0 E^2
Energy Stored in a Capacitor: U = 12 CV 2 Capacitance of a Capacitor with Dielectric: C = κC 0 Energy Stored in an Isolated Capacitor with Dielectric: U = 12 CV 2 = 12 κC 0 V 2 Dielectric Constant: κ = (^) CC 0 Induced Electric Field in a Dielectric: E⃗ induced = ⃗ E κ^0
Electrical Current
Average Electrical Current: I = ∆ ∆Qt Definition of an Ampere: 1 A = 1 C/s Electrical Current: I = ∆ ∆Qt Drift Velocity: vd = (^) nAeI Current Density: J⃗ = nevd Resistivity: ρ = (^) σ^1 Common Expression of Ohm’s Law: V = IR Resistivity as a Function of Temperature: ρ(T ) = ρ 0 [1 + α(T − T 0 )] Definition of Resistance: R = ρLA Resistance of a Cylinder of Material: R = ρLA Temperature Dependence of Resistance: R(T ) = R 0 [1 + α(T − T 0 )] Electric Power: P = IV = I^2 R = V^
2 R Power Dissipated by a Resistor: P = I^2 R = V^
2 R
Circuits with Capacitors and Resistors
Terminal Voltage of a Single Voltage Source: VT = V −IR Equivalent Resistance of a Series Circuit: Req = R 1 + R 2 + · · · + Rn Equivalent Resistance of a Parallel Circuit: 1 Req =^
1 R 1 +^
1 R 2 +^ · · ·^ +^
1 Rn Junction Rule:
P
Iin =
P
Iout Loop Rule:
P
∆V = 0
Terminal Voltage of N Voltage Sources in Series:P VT = Vi Terminal Voltage of N Voltage Sources in Parallel: VT = V 1 = V 2 = · · · = Vn Charge on a Charging Capacitor: Q(t) = Qmax
1 − e−t/RC^
Time Constant: τ = RC Current During Charging of a Capacitor: I(t) = VR e−t/RC Charge on a Discharging Capacitor: Q(t) = Q 0 e−t/RC Current During Discharging of a Capacitor: I(t) = − (^) RCQ^0 e−t/RC
Magnetic Forces and Fields
Force on a Charge in a Magnetic Field: F⃗ = q⃗v × B⃗ Magnitude of Magnetic Force: F = qvB sin θ Radius of a Particle’s Path in a Magnetic Field: r = mvqB Period of a Particle’s Motion in a Magnetic Field: T = (^2) qBπm Force on a Current-Carrying Wire in a Uniform Magnetic Field: F⃗ = Iℓ⃗B sin θ Magnetic Dipole Moment:μ⃗ = IAnˆ Torque on a Current Loop: τ =μ⃗ × B⃗ Energy of a Magnetic Dipole: U = −μ⃗ · B⃗
Drift Velocity in Crossed Electric and Magnetic Fields: v⃗ (^) d = EB Hall Potential: VH = (^) netIB Hall Potential in Terms of Drift Velocity: VH = E Bd Charge-to-Mass Ratio in a Mass Spectrometer: (^) mq = (^) B^22 Vr 2 Maximum Speed of a Particle in a Cyclotron: vmax = qBrm
Magnetic Fields and Magnetic Materi-
als
Permeability of Free Space: μ 0 = 4π × 10 −^7 T · m/A Contribution to Magnetic Field from a Current Element: d⃗B = μ 4 π^0 Id⃗lr× 2 ˆr Biot–Savart Law: B⃗ = μ 4 π^0
R (^) Id⃗l×rˆ r^2 Magnetic Field Due to a Long Straight Wire: B = μ 20 πrI Force Between Two Parallel Currents: F = μ^02 Iπr^1 I^2 L
Magnetic Field of a Current Loop: B = μ 20 RI
√^1
1+(z/R)^2
Amp`ere’s Law:
H
C
B^ ·^ d⃗l^ =^ μ^0 Ienc Magnetic Field Strength Inside a Solenoid: B = μ 0 nI Magnetic Field Strength Inside a Toroid: B = μ 20 πrN I Magnetic Permeability: μ = μ 0 μr Magnetic Field of a Solenoid Filled with Paramagnetic Material: B = μ 0 μr nI
Electromagnetic Induction
Magnetic Flux: ΦB =
R
S
B^ ·^ d⃗A Faraday’s Law: E = − dΦ dtB Motionally Induced Emf: E = BvL Motional Emf Around a Circuit: E = −
H ⃗
E · d⃗l Emf Produced by an Electric Generator: E = −N dΦ dtB
Inductance and Oscillations
Mutual Inductance by Flux: M = Φ I^211 Mutual Inductance in Circuits: M = N I^21 Φ^2 Self-Inductance in Terms of Magnetic Flux: L = Φ IB Self-Inductance in Terms of Emf: E = −L dIdt Self-Inductance of a Solenoid: L = μ 0 N^
(^2) A l Self-Inductance of a Toroid: L = μ^0 N^
(^2) A 2 πr Energy Stored in an Inductor: UL = 12 LI^2 Current as a Function of Time for a RL Circuit: I(t) = I 0
1 − e−^ τt^ Time Constant for a RL Circuit: τ = LR Charge Oscillation in LC Circuits: Q(t) = Q 0 cos(ωt) Angular Frequency in LC Circuits: ω = √^1 LC Current Oscillations in LC Circuits: I(t) = −I 0 sin(ωt) Charge as a Function of Time in an RLC Circuit: Q(t) = Q 0 e−^
t 2 τ (^) cos(ω′t) Angular Frequency in an RLC Circuit: ω′^ =
q 1 LC −^
1 4 R^2
AC Circuits
AC Voltage: V (t) = V 0 cos(ωt + ϕ) AC Current: I(t) = I 0 cos(ωt + ϕ)
Capacitive Reactance: XC = (^) ωC^1 RMS Voltage: Vrms = √V^02 RMS Current: Irms = √I^02 Inductive Reactance: XL = ωL Phase Angle of an RLC Series Circuit: tan(ϕ) = XL− RXC AC Version of Ohm’s Law: V = IZ Impedance of an RLC Series Circuit: Z =
p R^2 + (XL − XC )^2 Average Power Associated with a Circuit Element: P = V 0 I 0 2 cos^ ϕ Average Power Dissipated by a Resistor: P = I^2 rmsR Resonant Angular Frequency of a Circuit: ω 0 = √LC^1
Quality Factor of a Circuit: Q = (^) R^1
q L C Quality Factor of a Circuit in Terms of the Circuit Pa- rameters: Q = ωR^0 L Transformer Equation with Voltage: V V^12 = N N^12 Transformer Equation with Current: I I^12 = N N^21
Electromagnetic Waves
Displacement Current: Id = ϵ 0 ∂Φ ∂tE Gauss’s Law:
H
S
E^ ·^ d⃗A^ =^
Qenc ϵ 0 Gauss’s Law for Magnetism:
H
S
B^ ·^ d⃗A^ = 0 Faraday’s Law: E = − dΦ dtB Amp`ere-Maxwell Law:
H
C
B^ ·^ d⃗l^ =^ μ^0 Ienc^ +^ μ^0 ϵ^0
dΦE dt Wave Equation for Plane EM Wave: ∂
(^2) E⃗ ∂x^2 −^
1 c^2
∂^2 E⃗ ∂t^2 = 0 Speed of EM Waves: c = √μ^10 ϵ 0 Ratio of E Field to B Field in Electromagnetic Wave: E B =^ c Energy Flux (Poynting) Vector: S⃗ = (^) μ^10 E⃗ × B⃗ Average Intensity of an Electromagnetic Wave: I = 12 ϵ 0 cE 02 Radiation Pressure: P = Ic
Physical Constants
Atomic Mass Unit: u = 1. 66 × 10 −^27 kg Avogadro’s Number: NA = 6. 022 × 1023 mol−^1 Bohr Magneton: μB = 9. 27 × 10 −^24 J/T Bohr Radius: a 0 = 5. 29 × 10 −^11 m Boltzmann’s Constant: kB = 1. 38 × 10 −^23 J/K Compton Wavelength: λC = (^) mhec = 2. 426 × 10 −^12 m Coulomb Constant: ke = (^4) πϵ^10 = 8. 99 × 109 N · m^2 /C^2 Deuteron Mass: md = 3. 34 × 10 −^27 kg Electron Mass: me = 9. 11 × 10 −^31 kg Electron Volt: 1 eV = 1. 602 × 10 −^19 J Elementary Charge: e = 1. 60 × 10 −^19 C Gas Constant: R = 8.31 J/mol · K Gravitational Constant: G = 6. 674 × 10 −^11 N · m^2 /kg^2 Neutron Mass: mn = 1. 675 × 10 −^27 kg Nuclear Magneton: μN = 5. 05 × 10 −^27 J/T Permeability of Free Space: μ 0 = 4π × 10 −^7 T · m/A Permittivity of Free Space: ϵ 0 = 8. 85 × 10 −^12 C^2 /N · m^2 Planck’s Constant: h = 6. 626 × 10 −^34 J · s Proton Mass: mp = 1. 67 × 10 −^27 kg Rydberg Constant: R∞ = 1. 097 × 107 m−^1 Speed of Light in Vacuum: c = 3. 00 × 108 m/s
