PHYS 142 FORMULA SHEET Nicholas Salloum Electric Fields Electric Potential Coulomb’s Law: 𝑊 = ∫ 𝐹 ∙ 𝜕𝐿⃗ = ∫ 𝐹𝜕𝐿𝑐𝑜𝑠𝜙 = −Δ𝑈 𝜖0 = 8.85 × 10−12 𝑈= Test Charge in Electric Field: EPE of several point charges: 𝐹𝑜𝑛 𝑞 = 𝑞𝐸⃗ 𝑈= 1 𝑞 ∙ ∙ 𝑑̂ 4𝜋𝜖0 𝑑 2 Linear: 𝜆 = 𝑖=1 𝑉= 𝐿 Volume: 𝜌 = 𝑄 𝑉 𝑉= 1 𝑞 ∙ 4𝜋𝜖0 𝑑 2 1 𝑞 ∙ 4𝜋𝜖0 𝑑 2 1 𝜕𝑞 𝑉= ∫ 4𝜋𝜖0 𝑑 𝑏 1 𝜆 𝐸= ∙ 2𝜋𝜖0 𝑑 Electric Field inside infinite cylinder: 𝑉= 1 𝑞 ∙ 4𝜋𝜖0 𝑑 2 1 𝑄𝑑 ∙ 4𝜋𝜖0 𝑅 3 Electric Field of infinite charged sheet: 𝜎 2𝜖0 𝜎 𝜖0 𝜆 𝑅 𝑉= 𝑙𝑛 ( ) 2𝜋𝜖0 𝑑 Electric potential difference of infinite cylinder: 1 𝑄 𝑉= ∙ 4𝜋𝜖0 √(𝑑 2 + 𝑅 2 ) Electric potential difference along bisecting axis thin rod: 2 √𝑑 2 + (𝐿) 2 Electric Field along center bisector of disk: 𝑉= 𝜎 1 ∙ 1− 2 2𝜖0 √1 + (𝑅 ) ( 𝑑 ) Electric Field along center bisector of ring: 𝐸= 1 𝑄𝑑 ∙ 4𝜋𝜖0 √(𝑑 2 + 𝑅 2 )3 2 √(𝑑 2 + (𝐿 ) ) − (𝐿) 2 2 ) ( Electric potential difference between capacitor plates: 𝑉 = 𝐸𝑑 𝜕𝑉 𝜕𝑉 𝜕𝑉 𝐸⃗ = − ( 𝑖̂ + 𝑗̂ + 𝑘̂ ) 𝜕𝑥 𝜕𝑦 𝜕𝑧 ⃗𝑉 𝐸⃗ = −∇ 𝜖0 𝐸 2 2 Electric Flux Gauss’s Law: Φ𝐸 = ∮ 𝐸⃗ ∙ 𝜕𝐴 = ∮ 𝐸𝑐𝑜𝑠𝜙𝜕𝐴 = ∮ 𝐸⊥ 𝜕𝐴 = 𝑄 𝑙𝑛 4𝜋𝜖0 𝐿 2 √(𝑑 2 + (𝐿 ) ) + (𝐿) 2 2 Electric field as partial fractions of potential difference: Electric Energy Density in vacuum: 𝑢= 𝜆 𝑅 𝑙𝑛 ( ) 2𝜋𝜖0 𝑑 Electric potential difference along center bisector of ring: Electric Field along bisecting axis thin rod: 𝑄 1 𝑞 ∙ 4𝜋𝜖0 𝑅 Electric potential difference of infinite wire: 𝑉= Electric Field between capacitor plates: 1 𝑞 ∙ 4𝜋𝜖0 𝑑 Electric potential difference inside and at surface of conducting sphere: Electric Field inside insulating sphere: 𝐸= 𝑎 𝑉= 𝐸=0 Electric Field outside insulating sphere: 1 ∙ 4𝜋𝜖0 𝑏 Electric potential difference outside conducting sphere: Electric Field outside infinite cylinder: 𝐸= 𝑄𝑒𝑛𝑐𝑙 𝜖0 Radial electric field component: 𝜕𝑉 𝐸𝑅 = − 𝜕𝑅 Capacitance Gauss’s Law for defined shapes: 𝑄𝑒𝑛𝑐𝑙 Φ𝐸 = 𝐸𝐴𝑐𝑜𝑠𝜙 = 𝜖0 𝑛 1 1 1 1 1 =∑ = + + +⋯ 𝐶𝑒𝑞 𝐶𝑖 𝐶1 𝐶2 𝐶3 𝑛 𝐶𝑒𝑞 = ∑ 𝐶𝑖 = 𝐶1 + 𝐶2 + 𝐶3 + ⋯ 𝑖=1 Potential Energy stored in capacitor: 𝑈𝐶 = Definition of Capacitance: 𝐶= |𝑄| 𝑉 𝑄2 𝐶𝑉 2 𝑄𝑉 = = 2𝐶 2 2 Elec. Energy Density for capacitor: Electric potential difference from electric field: 𝑎 1 𝜆 𝐸= ∙ 2𝜋𝜖0 𝑑 𝐸= 1 𝑞𝑖 ∑ 4𝜋𝜖0 𝑑𝑖 ⃗ = ∫ 𝐸𝜕𝐿𝑐𝑜𝑠𝜙 𝑉𝑎 − 𝑉𝑏 = ∫ 𝐸⃗ ∙ 𝜕𝐿 Electric Field of infinite wire: 𝐸= Capacitors in series: 𝑖=1 𝑖=1 𝐸=0 𝐸= |𝑄| 2𝜋𝜖0 𝐿 = 𝑅 𝑉 𝑙𝑛 ( 𝑜𝑢𝑡 ) 𝑅𝑖𝑛 𝐶= Electric Potential cont. charge distribution Electric Field inside conducting sphere: 𝐸= Capacitance for coaxial cylinders: Capacitors in parallel: 𝑛 Electric Field outside conducting sphere: 𝐸= 1 𝑞 ∙ 4𝜋𝜖0 𝑑 Electric Potential several point charges: 𝐴 𝑄 Electric Field of point charge: 𝐸= 𝑞0 𝑞𝑖 ∑ 4𝜋𝜖0 𝑑𝑖 Electric Potential due to point charge: 𝑄 Surface: 𝜎 = |𝑄| 𝑅𝑖𝑛 𝑅𝑜𝑢𝑡 = 4𝜋𝜖0 ∙ 𝑉 𝑅𝑜𝑢𝑡 − 𝑅𝑖𝑛 𝑛 Electric Field Pt Charge: Charge density 1 𝑞1 𝑞2 ∙ 4𝜋𝜖0 𝑑 𝐴𝑝𝑙𝑎𝑡𝑒 |𝑄𝑝𝑙𝑎𝑡𝑒 | = 𝜖0 𝑉 𝑑 Capacitance for spherical concentric shells 𝐶= EPE of two point charges: Permittivity Constant: 𝐸⃗ = 𝐶= Work done by a force: 1 |𝑞1 ∙ 𝑞2 | ∙ 4𝜋𝜖0 𝑑2 𝐹𝑝𝑜𝑖𝑛𝑡 = Capacitance for parallel plate capacitor: 𝑢= 𝑈𝐶 𝜖0 𝐸 2 = 𝐴𝑑 2 Dielectric Constant (E0,C0,V0 in vacuum): 𝐾= 𝐶 𝑉0 𝐸0 = = 𝐶0 𝑉 𝐸 Permittivity of Dielectric: 𝜖 = 𝐾𝜖0 Charge per unit area on dielectric: 1 𝜎𝑖𝑛𝑑𝑢𝑐𝑒𝑑 = 𝜎 (1 − ) 𝐾 Electric Field between plates w/ dielectric: 𝐸= 𝜎 − 𝜎𝑖𝑛𝑑𝑖𝑐𝑒𝑑 𝜎 = 𝜖0 𝜖 Capacitance with dielectric: 𝐶 = 𝐾𝐶0 = 𝐾𝐴𝜖0 𝐴𝜖 = 𝑑 𝑑 Electric Energy Density with dielectric: 𝑢= 𝐾𝜖0 𝐸 2 𝜖𝐸 2 = 2 2 Current and Resistance Current through cross section: 𝜕𝑞 𝑖= = 𝑛|𝑞|𝑣𝑑 𝐴 𝜕𝑡 Current through unit cross section: 𝑖 𝐽 = = 𝑛|𝑞|𝑣𝑑 , 𝐽 = 𝑛|𝑞|𝑣𝑑 𝐴 Kirchhoff’s Junction Rule: ∑ 𝑖𝑖𝑛 = ∑ 𝑖𝑜𝑢𝑡 Resistivity of a material: 𝐸 𝑚𝑒 − 𝜌= = , 𝐸⃗ = 𝜌𝐽 𝐽 𝑛𝜏𝑞2 𝑒 − Conductivity of a material: 1 𝐽 𝜎= = 𝜌 𝐸 Resistivity of material at temperature: 𝜌(𝑇) = 𝜌0 [1 + 𝛼(𝑇 − 𝑇0)] Resistance of an ohmic conductor: 𝑉 𝜌𝐿 𝑅= = 𝑖 𝐴 Resistance of conductor at temperature 𝑅(𝑇) = 𝑅0 [1 + 𝛼(𝑇 − 𝑇0)] Resistors in series: 𝑛 𝑅𝑒𝑞 = ∑ 𝑅𝑖 = 𝑅1 + 𝑅2 + 𝑅3 + ⋯ 𝑖=1 Resistors in parallel: 𝑛 1 1 1 1 1 =∑ = + + +⋯ 𝑅𝑒𝑞 𝑅𝑖 𝑅1 𝑅2 𝑅3 𝑖=1 PHYS 142 FORMULA SHEET Nicholas Salloum Electromotive Force and Power EMF of ideal source: Magnetic Force 𝑅𝐻 = Magnetic Force on moving charge: 𝜀 = 𝑉𝑎𝑐𝑟𝑜𝑠𝑠 EMF of source with internal resistance: 𝑉 = 𝜀 − 𝑖𝑟 ⃗ ) , 𝐹 = |𝑞|𝑣𝐵𝑠𝑖𝑛𝜙 𝐹 = 𝑞(𝑣 × 𝐵 𝑉𝐻 𝐵 𝐵 = = 𝑖 𝑞𝑛𝑠ℎ𝑒𝑒𝑡 𝑞𝜌𝑑 ρ= charge density, d= width of sheet Magnetic Fields RHR for magnetic force on (+) charge: Current through unideal EMF source: Magnetic Constant: 𝜀 𝑅+𝑟 𝑖= Hall Resistance: 𝜇0 = 4𝜋 × 10−7 Magnetic Field of a moving charge: Net change in potential energy in loop: 𝜀 − 𝑖𝑟 − 𝑖𝑅 = 0 Power Definition: NOTE: if (-) charge, Force opposite thumb Magnetic Force on charge in electric field: ⃗) 𝐹 = 𝑞(𝐸⃗ + 𝑣 × 𝐵 𝑃 = 𝑉𝑖 Power input to an ohmic resistor: 𝑉2 𝑃 = 𝑉𝑖 = 𝑖 𝑅 = 𝑅 2 Power output of a source: 𝑃 = 𝑉𝑖 = 𝜀𝑖 − 𝑖 2 𝑅 ⃗ ×𝐵 ⃗ ) , 𝐹 = 𝐼𝐿𝐵𝑠𝑖𝑛𝜙 𝐹 = 𝐼(𝐿 ⃗L NOTE: if (-) charge, B opposite finger curl Magnetic Field of a current element: NOTE: ⃗L same direction as current Magnetic Force on non-straight wires: 𝑛 ⃗ ×𝐵 ⃗) 𝜕𝐹 = 𝐼(𝜕𝐿 ∆𝑉 = ∑ 𝑉𝑖 = 0 To be integrated according to shape Magnetic Force of parallel conductors: 𝑖=1 EMF & discharging C sign with Travel from - to +: 𝑇𝑟𝑎𝑣𝑒𝑙 → RHR for magnetic field moving (+)charge: RHR for magnetic force with current: 𝑃 = 𝑉𝑖 = 𝜀𝑖 + 𝑖 2 𝑅 Kirchhoff’s Loop Rule: 𝜇0 𝑞𝑣 × 𝑑̂ 𝜇0 |𝑞|𝑣𝑠𝑖𝑛𝜙 ∙ ,𝐵 = ∙ 4𝜋 𝑑2 4𝜋 𝑑2 Magnetic Force on conductor with current Power input to a source: Kirchhoff’s Loop Rule and Signs ⃗ = 𝐵 𝐹= ⃗ × 𝑑̂ 𝜇0 𝑖𝜕𝐿 ∙ 4𝜋 𝑑2 𝜇0 𝑖𝜕𝐿𝑠𝑖𝑛𝜙 𝜕𝐵 = ∙ 4𝜋 𝑑2 ⃗ = 𝜕𝐵 RHR for magnetic field of current: 𝜇0 𝐼1 𝐼2 𝐿 2𝜋𝑑 Relation between magnetic and electric F: +𝜀 EMF & discharging C sign with Travel from + to -: 𝑇𝑟𝑎𝑣𝑒𝑙 𝐹𝐵 𝑣2 = 𝜖0 𝜇0 𝑣 2 = 2 𝐹𝐸 𝑐 Magnetic Flux ← −𝜀 Definition of Magnetic Flux: R, L, charging C sign with Travel opposite current: 𝑇𝑟𝑎𝑣𝑒𝑙 → ← ⃗ ∙ 𝜕𝐴 = ∫ 𝐵𝑐𝑜𝑠𝜙𝜕𝐴 = ∫ 𝐵⊥ 𝜕𝐴 Φ𝐵 = ∫ 𝐵 ⃗ ∙ 𝜕𝐿 ⃗ = 𝜇0 𝑖𝑒𝑛𝑐𝑙𝑜𝑠𝑒𝑑 ∮𝐵 Magnetic Field of long straight conductor: Magnetic Flux w/ uniform B and A: Gauss’s Law for Magnetism (net flux=0): ⃗ ∙ 𝜕𝐴 = 0 ∮𝐵 +𝑉 R, L, charging C sign with Travel same as current: 𝑇𝑟𝑎𝑣𝑒𝑙 ← ← Magnetic Field aka magnetic flux density: 𝐵= 𝐶𝑢𝑟𝑟𝑒𝑛𝑡 𝜕Φ𝐵 𝜕𝐴⊥ Cyclotron −𝑉 Radius of motion of particle in cyclotron: RC Circuits 𝑅= 𝑚𝑣 |𝑞|𝐵 Angular speed in cyclotron: Time Constant: 𝜏 = 𝑅𝐶 Charging Capacitor charge equation: −𝑡 −𝑡 𝑞 = 𝜀𝐶 (1 − 𝑒 𝜏 ) = 𝑄𝑓 (1 − 𝑒 𝜏 ) Charging Capacitor voltage equation: −𝑡 𝑣 = 𝜀𝑓 (1 − 𝑒 𝜏 ) Charging Capacitor current equation: −𝑡 𝜕𝑞 𝜀 −𝑡 = 𝑒 𝜏 = 𝐼0 𝑒 𝜏 𝜕𝑡 𝑅 Discharging Capacitor charge equation: −𝑡 𝑞 = 𝑄0 𝑒 𝜏 𝜔= 𝑣 |𝑞|𝐵 = 𝑅 𝑚 Cyclotron frequency: 𝜔 1 = 2𝜋 𝑇 Applications of Motion of Particles 𝑓= Velocity selector for no particle deflection 𝑣= 𝐸 𝐵 Thomson’s Experiment q/m ratio: 𝑞𝑒 − 𝐸2 = 𝑚 2𝑉𝐵2 Discharging Capacitor current equation: −𝑡 𝑣 = 𝑉0 𝑒 𝜏 Discharging Capacitor current equation: −𝑡 𝜕𝑞 −𝑄0 −𝑡 𝑖= = 𝑒 𝜏 = 𝐼0 𝑒 𝜏 𝜕𝑡 𝜏 Hall Effect Hall Effect Equation: 𝑛𝑞 = −𝐽𝑥 𝐵𝑦 𝐸𝑧 𝜇0 𝑖 2𝜋𝑑 𝐵= Φ𝐵 = 𝐵𝐴𝑐𝑜𝑠𝜙 𝐶𝑢𝑟𝑟𝑒𝑛𝑡 𝑖= ⃗ same direction as current NOTE: L NOTE: B and I interchangeable here (solenoid)! Ampere’s Law: Magnetic Field at bisecting axis of loop: 𝐵= 𝜇0 𝑖𝑁𝑅 2 2√(𝑑 2 + 𝑅 2 )3 Magnetic Field at center of loop: 𝜇0 𝑖𝑁 2𝑅 𝐵= Magnetic Field inside cylinder conductor: 𝐵= 𝜇0 𝑖𝑑 2𝜋𝑅 2 Magnetic Field outside cylinder conductor 𝐵= 𝜇0 𝑖 2𝜋𝑑 Magnetic Field inside solenoid: 𝐵 = 𝜇0 𝑛𝑖 = 𝜇0 𝑁𝑖 𝐿 Magnetic Field outside solenoid: 𝐵=0 Magnetic Field between windings, toroid: 𝐵= 𝜇0 𝑖𝑁 2𝜋𝑑 Magnetic Field outside toroid: 𝐵=0 Enclosed current in solenoid and toroid: 𝑖𝑒𝑛𝑐𝑙𝑜𝑠𝑒𝑑 = 𝑛𝐿𝑖 = 𝑁𝑖 Magnetic Energy Density in vacuum: 𝑢= 𝐵2 2𝜇0 PHYS 142 FORMULA SHEET Nicholas Salloum Magnetic Materials Self-induced emf: 𝜀 = −𝐿 Planck’s Constant: ℎ = 6.626 × 10−34 Mutual Inductance: Bohr Magneton: ℎ𝑞𝑒 − 𝜇𝐵 = = 9.274 × 10−24 4𝜋𝑚𝑒 − 𝑀= 𝜇 = 𝐾𝑚 𝜇0 𝜀1 = −𝑀 𝜒𝑚 = 𝐾𝑚 − 1 Magnetization with temperature: 𝑀=𝐶 𝐵 𝑇 𝜕𝑖2 𝜕𝑡 , 𝜀2 = −𝑀 𝜕𝑖1 𝜕𝑡 1 𝑅2 𝜔′ = √ − 2 𝐿𝐶 4𝐿 𝐿𝑖 2 4𝐿 NOTE: only true for 𝑅2 < 𝐶 Energy stored in inductor over time: 𝑈= −𝑡 𝐿𝑖0 2 −𝑡 𝑒 𝜏 = 𝑈0 𝑒 𝜏 2 NOTE: τ depends on circuit (RL, RC, RLC) Inductors in series: Induction Kirchhoff’s Rule for RLC circuit (ODE): 𝜕 2 𝑞 𝑅 𝜕𝑞 1 + + 𝑞=0 𝜕𝑡 2 𝐿 𝜕𝑡 𝐿𝐶 Angular Frequency RLC (underdamped): 2 2 𝐵 𝑢= 2𝜇 RLC Circuits Energy stored in inductor: 𝑈= Where C is Curie’s constant Magnetic Energy Density in material: Instantaneous charge equation for RLC: 𝑅 𝜕Φ𝐵 𝜕𝑡 𝐿𝑒𝑞 = ∑ 𝐿𝑖 = 𝐿1 + 𝐿2 + 𝐿3 + ⋯ 𝑖=1 NOTE: this is the solution to the ODE of RLC circuits (see above) AC Circuits Sinusoidal voltage: 𝑣 = 𝑉cos(𝜔𝑡) Inductors in parallel: RHR for direction of positive emf: 𝑛 Sinusoidal current: 𝑖=1 Rectified average current: 1 1 1 1 1 =∑ = + + +⋯ 𝐿𝑒𝑞 𝐿𝑖 𝐿1 𝐿2 𝐿3 𝑖 = 𝐼cos(𝜔𝑡) RL Circuits 𝐼𝑟𝑎𝑣 = NOTE: use RHR for magnetic field of current is to determine direction of induced emf. The induced magnetic field would be directed to oppose the change in external magnetic field. That is, an increase in external magnetic field creates an induced magnetic field directed in a way to decrease net magnetic field. Similarly, a decrease in external magnetic field creates an induced magnetic field directed in a way to increase net magnetic field. 𝐿 𝜏= 𝑅 𝐼𝑟𝑚𝑠 = −𝑡 −𝑡 𝜀 (1 − 𝑒 𝜏 ) = 𝐼𝑓 (1 − 𝑒 𝜏 ) 𝑅 Rate of current growth in RL circuits: 𝜕𝑖 𝜀 −𝑡 = 𝑒𝜏 𝜕𝑡 𝐿 Power given by emf source in RL circuit: 𝜕𝑖 𝜕𝑡 Current Decay in RL circuit: ⃗ ) ∙ 𝜕𝐿 ⃗ 𝜀 = ∮(𝑣 × 𝐵 Energy in RL circuit with no emf source: Motional emf for conductor length in BꞱ: 0 = 𝑖 2 𝑅 + 𝐿𝑖 𝜀 = 𝑣𝐵𝐿 𝜕𝑖 𝜕𝑡 LC Circuits RHR for direction of emf in motional emf: 𝑉 √2 Average power into any AC circuit: 𝑉𝐼𝑐𝑜𝑠𝜙 = 𝑉𝑟𝑚𝑠 𝐼𝑟𝑚𝑠 𝑐𝑜𝑠𝜙 2 Instantaneous power delivered by source: 𝑝 = 𝑣𝑖 Angular frequency at resonance: 𝜔0 = −𝑡 𝑖 = 𝐼0 𝑒 𝜏 Motional induced emf general equation: 𝑉𝑟𝑚𝑠 = 𝑃𝑎𝑣𝑔 = 𝑃 = 𝜀𝑖 = 𝑖 2 𝑅 + 𝐿𝑖 𝐼 √2 RMS Voltage: Current growth in RL circuits: 𝑖= 2 𝐼 𝜋 RMS Current: Time Constant: NOTE: for negative emf, opposite curling Lenz’s Law: The direction of any magnetic induction effect (ε,i) is such as to oppose the cause of the effect. )𝑡 𝑞 = 𝐴𝑒 −(2𝐿 cos(𝜔′ 𝑡 + 𝜙) 𝑛 Faraday’s Law (induced emf): 𝜀 = −𝑁 𝑁2 Φ𝐵2 𝑁1 Φ𝐵1 = 𝑖1 𝑖2 Mutually induced emfs: Permeability of magnetic material: Magnetic Susceptibility: 𝜕𝑖 𝜕𝑡 1 √𝐿𝐶 , 𝑓0 = 𝜔0 2𝜋 Elements in AC Circuits Instantaneous voltage across resistor: Kirchhoff’s Rule for LC circuit (ODE): 2 𝜕 𝑞 1 + 𝑞=0 𝜕𝑡 2 𝐿𝐶 Angular Frequency for LC circuit: 1 𝜔=√ 𝐿𝐶 Instantaneous charge equation for LC: 𝑞 = 𝑄𝑚𝑎𝑥 cos(𝜔𝑡 + 𝜙) Power generated in motional emf: 𝑃 = 𝐹𝑣 = (𝐵𝐿𝑣)2 𝑅 Induced Electric Field: 𝐸= 1 𝜕Φ𝐵 ∙| | 2𝜋𝑑 𝜕𝑡 Faraday’s Law for a stationary path: ⃗ =− 𝜀 = ∮ 𝐸⃗ ∙ 𝜕𝐿 Inductance 𝑖= 𝜕𝑞 = −𝜔𝑄𝑚𝑎𝑥 sin(𝜔𝑡 + 𝜙) 𝜕𝑡 𝑖 = ±𝜔√𝑄𝑚𝑎𝑥 2 − 𝑞2 Power given by capacitor to circuit: 𝑄2 𝐿𝑖 2 𝑞 2 = + 2𝐶 2 2𝐶 Frequency of oscillation in LC circuit: Self-Inductance: 𝐿= 𝜕Φ𝐵 𝜕𝑡 NOTE: this is the solution to the ODE of LC circuits (see above) Instantaneous current equations: 𝑁Φ𝐵 𝑖 𝑓= 𝜔 1 = 2𝜋 𝑇 𝑣𝑅 = 𝑖𝑅 = 𝐼𝑅cos(𝜔𝑡) = 𝑉𝑅 cos(𝜔𝑡) NOTE: In AC circuit with R only, voltage phasor is in phase with current phasor Voltage amplitude across resistor: 𝑉𝑅 = 𝐼𝑅 RMS Voltage amplitude across resistor: 𝑉𝑅 𝑟𝑚𝑠 = 𝐼𝑟𝑚𝑠 𝑅 Average power into a resistor: 𝑃𝑅 𝑎𝑣𝑔 = 𝐼𝑉 𝑉𝑟𝑚𝑠 2 = 𝐼𝑟𝑚𝑠 2 𝑅 = = 𝑉𝑟𝑚𝑠 𝐼𝑟𝑚𝑠 2 𝑅 Instantaneous voltage across inductor: 𝜕𝑖 = −𝐼𝜔𝐿𝑠𝑖𝑛(𝜔𝑡) 𝜕𝑡 𝑣𝐿 = −𝑉𝐿 𝑠𝑖𝑛(𝜔𝑡) 𝜋 𝑣𝐿 = 𝐼𝜔𝐿𝑐𝑜𝑠 (𝜔𝑡 + ) 2 𝜋 𝑣𝐿 = 𝑉𝐿 𝑐𝑜𝑠 (𝜔𝑡 + ) 2 𝑣𝐿 = 𝐿 NOTE: In AC circuit with L only, voltage phasor leads ahead of current phasor by 90° Inductive Reactance: 𝑋𝐿 = 𝜔𝐿 Voltage amplitude across inductor: 𝑉𝐿 = 𝐼𝑋𝐿 = 𝐼𝜔𝐿 PHYS 142 FORMULA SHEET Nicholas Salloum RMS Voltage amplitude across inductor: 𝑉𝐿 𝑟𝑚𝑠 = 𝐼𝑟𝑚𝑠 𝑋𝐿 Average power into an inductor: 𝑃𝐿 𝑎𝑣𝑔 = 0 Instantaneous charge in capacitor: 𝑞= 𝐼 𝑠𝑖𝑛(𝜔𝑡) 𝜔 𝜕Φ𝐵 𝜕𝑡 𝜕Φ𝐸 ⃗ ⃗ ∮ 𝐵 ∙ 𝜕𝐿 = 𝜇0 (𝑖𝐶 + 𝜖0 ) 𝜕𝑡 𝑒𝑛𝑐𝑙𝑜𝑠𝑒𝑑 ⃗ =− ∮ 𝐸⃗ ∙ 𝜕𝐿 Relation between magnitudes EF and MF: 𝐸 = 𝑐𝐵 𝐵 = 𝜖0 𝜇0 𝑐𝐸 𝐼 𝑠𝑖𝑛(𝜔𝑡) 𝜔𝐶 𝑣𝐶 = 𝑉𝐶 𝑠𝑖𝑛(𝜔𝑡) 𝐼 𝜋 𝑣𝐶 = 𝑐𝑜𝑠 (𝜔𝑡 − ) 𝜔𝐶 2 𝜋 𝑣𝐶 = 𝑉𝐶 𝑐𝑜𝑠 (𝜔𝑡 − ) 2 NOTE: In AC circuit with C only, voltage phasor lags behind current phasor by 90° Capacitive Reactance: 𝑋𝐶 = 1 𝜔𝐶 Wave equation for EMR: 𝜕 2 𝐸𝑦 (𝑥, 𝑡) 𝜕 2 𝐸𝑦 (𝑥, 𝑡) = 𝜖0 𝜇0 2 𝜕𝑥 𝜕𝑡 2 Speed of light in vacuum: 1 𝑐= = 𝜆𝑓 = 3 × 108 √𝜖0 𝜇0 Speed of EM wave in medium: 𝑣 = 𝜆𝑓 Sinusoidal wave equations: 𝐸⃗ (𝑥, 𝑡) = [𝐸𝑚𝑎𝑥𝑐𝑜𝑠(𝑘𝑥 ± 𝜔𝑡)]𝑗̂ ̂ ⃗ (𝑥, 𝑡) = [𝐵𝑚𝑎𝑥𝑐𝑜𝑠(𝑘𝑥 ± 𝜔𝑡)]𝑘 𝐵 𝐸𝑚𝑎𝑥 = 𝑐𝐵𝑚𝑎𝑥 Voltage amplitude across capacitor: 𝐼 𝜔𝐶 RMS Voltage amplitude across capacitor: NOTE: (+) when EMR moving in −𝑖̂ , and (-) when moving in +𝑖̂ Wave number: 𝑉𝐶 𝑟𝑚𝑠 = 𝐼𝑟𝑚𝑠 𝑋𝐶 𝑘= 𝑉𝐶 = 𝐼𝑋𝐶 = Average power into a capacitor: 𝑃𝐶 𝑎𝑣𝑔 = 0 Angular frequency: 𝜔 = 2𝜋𝑓 = RLC AC Circuits Impedance of AC circuit: 𝑍 = √𝑅 2 + (𝑋𝐿 − 𝑋𝐶 )2 1 2 𝑍 = √𝑅 2 + [𝜔𝐿 − ( )] 𝜔𝐶 Instantaneous voltage expression: 𝑣 = 𝑉cos(𝜔𝑡 + 𝜙) Energy density in EMR wave: 2 𝑢= 𝑆= 𝑉 = 𝐼𝑍 𝑡𝑎𝑛𝜙 = 2 𝐵 𝜖0 𝐸 + = 𝜖0 𝐸 2 2𝜇0 2 Energy flow per unit time per unit area: RMS Voltage amplitude in RLC: 𝑉𝑟𝑚𝑠 = 𝐼𝑟𝑚𝑠 𝑍 1 𝜕𝑈 𝐸𝐵 = 𝜖0 𝑐𝐸 2 = 𝐴 𝜕𝑡 𝜇0 1 ⃗ 𝑆 = 𝐸⃗ × 𝐵 𝜇0 RHR Poynting Vector 𝑆, direction of EMR: 𝑉𝐿 − 𝑉𝐶 𝑋𝐿 − 𝑋𝐶 = 𝑉𝑅 𝑅 ⃗S Phasor analysis for RLC AC circuits 𝑿𝑳 > 𝑿𝑪 (𝑽𝑳 > 𝑽𝑪 ) 𝑿𝑳 < 𝑿𝑪 (𝑽𝑳 < 𝑽𝑪 ) Voltage leads 𝑋𝐿 − 𝑋𝐶 > 0 𝑡𝑎𝑛𝜙 > 0 0 < 𝜙 < 90° Voltage lags 𝑋𝐿 − 𝑋𝐶 < 0 𝑡𝑎𝑛𝜙 < 0 −90° < 𝜙 < 0 Transformers Transformer equation, unknown current: 𝑉2 𝑁2 = 𝑉1 𝑁1 Transformer equation, unknown loops: 𝑉1 𝐼1 = 𝑉2 𝐼2 Combining above 2 equations: 𝑉1 𝑅 = 𝐼1 𝑁2 2 ( ) 𝑁1 2𝜋 = 𝑐𝑘 𝑇 Speed of EMR in dielectric: 1 1 1 𝑐 𝑣= =( )= √𝜖𝜇 √𝐾𝐾𝑚 √𝜖0𝜇0 √𝐾𝐾𝑚 Voltage amplitude in RLC: Phase angle in RLC: 2𝜋 𝜆 Maxwell’s Equations: 𝑄𝑒𝑛𝑐𝑙 ∮ 𝐸⃗ ∙ 𝜕𝐴 = 𝜖0 ⃗ ∙ 𝜕𝐴 = 0 ∮𝐵 𝐸𝑦 (𝑥, 𝑡) = −2𝐸𝑚𝑎𝑥 𝑠𝑖𝑛(𝑘𝑥)𝑠𝑖𝑛(𝜔𝑡) 𝐵𝑧 (𝑥, 𝑡) = −2𝐵𝑚𝑎𝑥 𝑐𝑜𝑠(𝑘𝑥)𝑐𝑜𝑠(𝜔𝑡) Nodal points of Electric Field Wave: 𝜆 3𝜆 𝑥 = 0, , 𝜆, , … 2 2 NOTE: these are the anti-nodal points of the magnetic field wave Nodal points of Magnetic Field Wave: 𝜆 3𝜆 5𝜆 𝑥= , , … 4 4 4 NOTE: these are the anti-nodal points of the electric field wave Wavelengths of standing waves in cavity: 𝜆𝑛 = 2𝐿 , (𝑛 = 1, 2, 3, … ) 𝑛 Frequencies of standing waves in cavity: 𝑓𝑛 = 𝑐 𝑐𝑛 = , (𝑛 = 1, 2, 3, … ) 𝜆𝑛 2𝐿 Optics Speed of light: 𝑐 = 3.00 × 108 Index of refraction of material: 𝑛= 𝑐 𝑣 Law of reflection: 𝜙𝑖𝑛𝑐𝑖𝑑𝑒𝑛𝑐𝑒 = 𝜙𝑟𝑒𝑓𝑙𝑒𝑐𝑡𝑖𝑜𝑛 Law of refraction: 𝑛1 𝑠𝑖𝑛𝜙𝑖𝑛𝑐𝑖𝑑𝑒𝑛𝑐𝑒 = 𝑛2 𝑠𝑖𝑛𝜙𝑟𝑒𝑓𝑟𝑎𝑐𝑡𝑖𝑜𝑛 Wavelength change in material: 𝜆𝑓 = 𝜆𝑖 𝑛 NOTE: frequency remains the same upon transmission Total internal reflection: 𝑛2 𝜙𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙 = 𝑎𝑟𝑐𝑠𝑖𝑛 ( ) 𝑛1 Measure of light dispersion: 𝜙𝑑𝑖𝑠𝑝𝑒𝑟𝑠𝑖𝑜𝑛 = |𝜙𝜆1 − 𝜙𝜆2 | Intensity unpolarized light hitting polarizer: ⃗ E 𝐼= 𝐼0 2 Intensity polarized light hitting polarizer: ⃗ B 𝐼 = 𝐼0 𝑐𝑜𝑠 2 𝜙 NOTE: this is the same direction of EMR Total energy flow per unit time: 𝑃 = ∮ 𝑆 ∙ 𝜕𝐴 Brewster Law: 𝑛2 𝜙𝑝𝑜𝑙𝑎𝑟𝑖𝑧𝑎𝑡𝑖𝑜𝑛 = 𝑎𝑟𝑐𝑡𝑎𝑛 ( ) 𝑛1 Geometric Optics Intensity of radiation: 𝐼= 𝐸𝑚𝑎𝑥 𝐵𝑚𝑎𝑥 𝐸𝑚𝑎𝑥 2 𝜖0 𝑐𝐸𝑚𝑎𝑥 2 = = 2𝜇0 2𝜇0 𝑐 2 Momentum carried per unit volume: 𝜕𝑝 𝐸𝐵 𝑆 = = 𝜕𝑉 𝜇0 𝑐 2 𝑐 2 Momentum flow rate: 1 𝜕𝑝 𝑆 𝐸𝐵 = = 𝐴 𝜕𝑡 𝑐 𝜇0 𝑐 Average momentum flow rate: Electromagnetic Waves 2𝐼 𝑐 Sinusoidal standing wave equations: Magnetic field magnitude: Instantaneous voltage across capacitor: 𝑣𝐶 = 𝑝𝑟𝑎𝑑 = 1 𝜕𝑝 𝐼 ( ) = 𝐴 𝜕𝑡 𝑎𝑣𝑔 𝑐 Radiation pressure, EMR wave absorbed: 𝑝𝑟𝑎𝑑 = 𝐼 𝑐 Radiation pressure, EMR wave reflected: Sign Rules for geometric optics: -Object distance s: when the object is the same side as incoming light, (+). Otherwise, (-). -Image distance s’: when the image is the same side as outgoing light, (+). Otherwise, (-). -Radius of spherical surface R: when center is the same side as outgoing light, (+). Otherwise, (-). Virtual and Real images: Real Image when outgoing rays pass through image point. Otherwise, Virtual Image formation by plane mirror: 𝑠 = −𝑠′ Lateral magnification, plane mirror: 𝑖𝑚𝑎𝑔𝑒 ℎ𝑒𝑖𝑔ℎ𝑡 𝑚= =1 𝑜𝑏𝑗𝑒𝑐𝑡 ℎ𝑒𝑖𝑔ℎ𝑡 PHYS 142 FORMULA SHEET Nicholas Salloum Object image relation, spherical mirror: 1 1 2 1 + = = 𝑠 𝑠′ 𝑅 𝑓 Lateral magnification, spherical mirror: 𝑖𝑚𝑎𝑔𝑒 ℎ𝑒𝑖𝑔ℎ𝑡 𝑠′ 𝑚= =− 𝑜𝑏𝑗𝑒𝑐𝑡 ℎ𝑒𝑖𝑔ℎ𝑡 𝑠 Object image relation, spherical refracting surface: 𝑛1 𝑛2 𝑛2 − 𝑛1 + = 𝑠 𝑠′ 𝑅 Lateral magnification, spherical refracting surface: 𝑚= 𝑖𝑚𝑎𝑔𝑒 ℎ𝑒𝑖𝑔ℎ𝑡 𝑛1 𝑠′ =− 𝑜𝑏𝑗𝑒𝑐𝑡 ℎ𝑒𝑖𝑔ℎ𝑡 𝑛2 𝑠 Object image relation, plane refracting surface: 𝑛1 𝑛2 + =0 𝑠 𝑠′ Lateral magnification, plane refracting surface: 𝑚= 𝑖𝑚𝑎𝑔𝑒 ℎ𝑒𝑖𝑔ℎ𝑡 =1 𝑜𝑏𝑗𝑒𝑐𝑡 ℎ𝑒𝑖𝑔ℎ𝑡 Thin lens focal length equation: 1 1 1 + = 𝑠 𝑠′ 𝑓 NOTE: F1=F2 Thin lens Lateral magnification: 𝑚= 𝑖𝑚𝑎𝑔𝑒 ℎ𝑒𝑖𝑔ℎ𝑡 𝑠′ =− 𝑜𝑏𝑗𝑒𝑐𝑡 ℎ𝑒𝑖𝑔ℎ𝑡 𝑠 Lens-maker equation: 1 1 1 = (𝑛 − 1) ( − ) 𝑓 𝑅1 𝑅2 Camera f-number: 𝑓 𝑛𝑢𝑚𝑏𝑒𝑟 = 𝑓 𝐷 Where D is the diameter of the aperture
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