phys142 ultimate formula sheet
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:
⃗ ) , 𝐹 = |𝑞|𝑣𝐵𝑠𝑖𝑛𝜙 𝐹 = 𝑞(𝑣 × 𝐵
𝜀 = 𝑉𝑎𝑐𝑟𝑜𝑠𝑠 EMF of source with internal resistance:
𝑉 = 𝜀 − 𝑖𝑟
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
NOTE: if (-) charge, Force opposite thumb Magnetic Force on charge in electric field:
Power Definition:
⃗) 𝐹 = 𝑞(𝐸⃗ + 𝑣 × 𝐵
𝑃 = 𝑉𝑖 Power input to an ohmic resistor:
⃗ = 𝐵
𝜇0 𝑞𝑣 × 𝑑̂ 𝜇0 |𝑞|𝑣𝑠𝑖𝑛𝜙 ∙ ,𝐵 = ∙ 4𝜋 𝑑2 4𝜋 𝑑2
RHR for magnetic field moving (+)charge:
Magnetic Force on conductor with current
𝑉2 𝑃 = 𝑉𝑖 = 𝑖 𝑅 = 𝑅
⃗ ×𝐵 ⃗ ) , 𝐹 = 𝐼𝐿𝐵𝑠𝑖𝑛𝜙 𝐹 = 𝐼(𝐿
2
RHR for magnetic force with current:
Power output of a source:
𝑃 = 𝑉𝑖 = 𝜀𝑖 − 𝑖 2 𝑅
⃗L
NOTE: if (-) charge, B opposite finger curl Magnetic Field of a current element:
Power input to a source:
𝑃 = 𝑉𝑖 = 𝜀𝑖 + 𝑖 2 𝑅
Kirchhoff’s Loop Rule and Signs Kirchhoff’s Loop Rule:
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
→
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
⃗ ∙ 𝜕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:
= 𝑄𝑓 (1 −
−𝑡 𝑒𝜏)
Charging Capacitor voltage equation: −𝑡
𝑣 = 𝜀𝑓 (1 − 𝑒 𝜏 ) Charging Capacitor current equation:
𝑖=
Magnetic Field of long straight conductor:
−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 Gauss’s Law for Magnetism (net flux=0):
𝑞 = 𝜀𝐶 (1 −
⃗ ∙ 𝜕𝐿 ⃗ = 𝜇0 𝑖𝑒𝑛𝑐𝑙𝑜𝑠𝑒𝑑 ∮𝐵
Magnetic Flux w/ uniform B and A:
𝐶𝑢𝑟𝑟𝑒𝑛𝑡
−𝑡 𝑒𝜏)
⃗ 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
,
𝐵2 𝑢= 2𝜇
RLC Circuits
𝜀2 = −𝑀
𝜕𝑖1 𝜕𝑡
1 𝑅2 𝜔′ = √ − 2 𝐿𝐶 4𝐿
2
𝐿𝑖 2
NOTE: only true for 𝑅2
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:
𝑡𝑎𝑛𝜙 =
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:
𝑋𝐶 =
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
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:
⃗ ) , 𝐹 = |𝑞|𝑣𝐵𝑠𝑖𝑛𝜙 𝐹 = 𝑞(𝑣 × 𝐵
𝜀 = 𝑉𝑎𝑐𝑟𝑜𝑠𝑠 EMF of source with internal resistance:
𝑉 = 𝜀 − 𝑖𝑟
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
NOTE: if (-) charge, Force opposite thumb Magnetic Force on charge in electric field:
Power Definition:
⃗) 𝐹 = 𝑞(𝐸⃗ + 𝑣 × 𝐵
𝑃 = 𝑉𝑖 Power input to an ohmic resistor:
⃗ = 𝐵
𝜇0 𝑞𝑣 × 𝑑̂ 𝜇0 |𝑞|𝑣𝑠𝑖𝑛𝜙 ∙ ,𝐵 = ∙ 4𝜋 𝑑2 4𝜋 𝑑2
RHR for magnetic field moving (+)charge:
Magnetic Force on conductor with current
𝑉2 𝑃 = 𝑉𝑖 = 𝑖 𝑅 = 𝑅
⃗ ×𝐵 ⃗ ) , 𝐹 = 𝐼𝐿𝐵𝑠𝑖𝑛𝜙 𝐹 = 𝐼(𝐿
2
RHR for magnetic force with current:
Power output of a source:
𝑃 = 𝑉𝑖 = 𝜀𝑖 − 𝑖 2 𝑅
⃗L
NOTE: if (-) charge, B opposite finger curl Magnetic Field of a current element:
Power input to a source:
𝑃 = 𝑉𝑖 = 𝜀𝑖 + 𝑖 2 𝑅
Kirchhoff’s Loop Rule and Signs Kirchhoff’s Loop Rule:
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
→
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
⃗ ∙ 𝜕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:
= 𝑄𝑓 (1 −
−𝑡 𝑒𝜏)
Charging Capacitor voltage equation: −𝑡
𝑣 = 𝜀𝑓 (1 − 𝑒 𝜏 ) Charging Capacitor current equation:
𝑖=
Magnetic Field of long straight conductor:
−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 Gauss’s Law for Magnetism (net flux=0):
𝑞 = 𝜀𝐶 (1 −
⃗ ∙ 𝜕𝐿 ⃗ = 𝜇0 𝑖𝑒𝑛𝑐𝑙𝑜𝑠𝑒𝑑 ∮𝐵
Magnetic Flux w/ uniform B and A:
𝐶𝑢𝑟𝑟𝑒𝑛𝑡
−𝑡 𝑒𝜏)
⃗ 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
,
𝐵2 𝑢= 2𝜇
RLC Circuits
𝜀2 = −𝑀
𝜕𝑖1 𝜕𝑡
1 𝑅2 𝜔′ = √ − 2 𝐿𝐶 4𝐿
2
𝐿𝑖 2
NOTE: only true for 𝑅2
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:
𝑡𝑎𝑛𝜙 =
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:
𝑋𝐶 =
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
