Scalar Product (Dot Product) Vector Product (Cross Product) ⃗ ⋅𝑩 ⃗⃗ = 𝐴𝐵 cos 𝜙 𝑨 ⃗ ×𝑩 ⃗⃗ | = 𝐴𝐵 sin 𝜙 and apply the right-hand rule |𝑨 ⃗ ⋅𝑩 ⃗⃗ = 𝐴𝑥 𝐵𝑥 + 𝐴𝑦 𝐵𝑦 + 𝐴𝑧 𝐵𝑧 𝑨 ̂ ⃗ ×𝑩 ⃗⃗ = (𝐴𝑦 𝐵𝑧 − 𝐴𝑧 𝐵𝑦 )𝒊̂ + (𝐴𝑧 𝐵𝑥 − 𝐴𝑥 𝐵𝑧 )𝒋̂ + (𝐴𝑥 𝐵𝑦 − 𝐴𝑦 𝐵𝑥 )𝒌 𝑨 Electric Flux and Gauss’s Law ⃗𝑬 of Symm. Q Distrib. ⃗𝑬 and Coulomb’s Law ⃗ Φ𝐸 = ∫ ⃗𝑬 ⋅ 𝑑𝑨 ⃗𝑬 = 1 4𝜋𝜖 𝑞 ̂ 2𝒓 𝑟 0 ⃗ = 1 𝑬 2𝜋𝜖 0 ⃗ = ⃗ ⋅ 𝑑𝑨 Φ𝐸 = ∮ 𝑬 𝑄𝑒𝑛𝑐𝑙 𝜖0 ⃗ ⃗𝑬 = 𝑭 𝑞 ⃗ ⃗ = 𝑞𝒅 𝒑 0 𝜆 𝒓̂ 𝑟 ⃗ = 1 𝑭 4𝜋𝜖 ⃗𝑬 = 𝜎 𝒏 ̂ 2𝜖 ⃗ =𝒏 ̂ 𝑑𝐴 𝑑𝑨 Electric Dipoles 0 1 𝐹 = 4𝜋𝜖 0 0 𝑞𝑞0 𝒓̂ 𝑟2 ⃗ ⃗ =𝒑 ⃗ ×𝑬 𝝉 |𝑞1 𝑞2 | ⃗ ⋅ ⃗𝑬 𝑈 = −𝒑 𝑟2 Electric Potential ↔ Electric Field Electric Potential Energy Electric Potential 𝑊𝑎→𝑏 = −∆𝑈 = 𝑈𝑎 − 𝑈𝑏 𝑉= 𝑈 𝑞0 𝑏 𝑉𝑎 − 𝑉𝑏 = ∫𝑎 ⃗𝑬 ⋅ 𝑑𝒍 𝑉= 1 𝑞 4𝜋𝜖0 𝑟 𝐸𝑥 = − 𝑈= 1 𝑞𝑞0 4𝜋𝜖0 𝑟 𝑞 𝑞 1 𝑞 𝜕𝑉 𝜕𝑥 ; 𝐸𝑦 = − 𝜕𝑉 𝜕𝑦 ; 𝐸𝑧 = − 𝑉 = 4𝜋𝜖 ∑𝑖 𝑟𝑖 Capacitance Equivalent Capacitance Energy in a Capacitor 1 𝐶𝑒𝑞 𝑈 = 2𝐶 = 2 𝐶𝑉 2 = 2 𝑄𝑉 𝑖 0 𝑄 𝐶=𝑉 𝑎𝑏 𝐶 = 𝜖0 𝐴 𝑑 or 𝐶 = 𝐾𝜖0 𝐴 𝑑 𝑖 1 1 1 2 = 𝐶 +𝐶 +⋯ 𝐶𝑒𝑞 = 𝐶1 + 𝐶2 + ⋯ 𝜕𝑉 𝜕𝑧 ̂ 𝜕𝑉) ⃗𝑬 = − (𝒊̂ 𝜕𝑉 + 𝒋̂ 𝜕𝑉 + 𝒌 𝜕𝑥 𝜕𝑦 𝜕𝑧 𝑈 = 4𝜋𝜖0 ∑𝑖 𝑟𝑖 0 (− to +) 𝑄2 1 1 2 1 1 2 𝑢 = 𝜖0 𝐸 2 or 𝑢 = 𝐾𝜖0 𝐸 2 When ⃗𝑬 is in a polarizable material, 𝜖0 is replaced in all equations with 𝐾𝜖0 (𝐾 is the dielectric constant). Current and Current Density 𝐼= 𝑑𝑄 𝑑𝑡 = 𝑛|𝑞|𝑣𝑑 𝐴 𝐽= 𝐼 𝐴 (𝐼𝑓 𝐽(𝑟), 𝐽(𝑟) = Resistivity and Resistance 𝜌= 𝑑𝐼 ) 𝑑𝐴 𝐸 𝐽 Battery, Ohm’s Law and Power 𝑉𝑎𝑏 = ℰ − 𝐼𝑟 𝜌(𝑇) = 𝜌0 [1 + 𝛼(𝑇 − 𝑇0 )] 𝜌𝐿 𝐴 𝐼= 𝑉𝑎𝑏 𝑅 or 𝑉𝑎𝑏 = 𝐼𝑅 𝑃 = 𝑉𝑎𝑏 𝐼 ; 𝑃 = 𝐼 2 𝑅 = 2 𝑉𝑎𝑏 𝑅 ⃗𝑑 𝑱 = 𝑛𝑞𝒗 𝑅= Kirchhoff’s Rules Equivalent Resistance R-C Circuits ∑ 𝐼 = 0 (𝑗𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑟𝑢𝑙𝑒) 𝑅𝑒𝑞 = 𝑅1 + 𝑅2 + ⋯ 𝜏 = 𝑅𝐶 ∑ 𝑉 = 0 (𝑙𝑜𝑜𝑝 𝑟𝑢𝑙𝑒) 1 𝑅𝑒𝑞 𝑞 = 𝑄𝑓 (1 − 𝑒 −𝑡/𝜏 ) ; 𝑄𝑓 = 𝐶ℰ 1 1 =𝑅 +𝑅 +⋯ 1 2 𝑞 = 𝑄0 𝑒 −𝑡/𝜏 𝑖= 𝑑𝑞 𝑑𝑡 = 𝐼0 𝑒 −𝑡/𝜏 Magnetic Flux and Gauss’s Law Magnetic Force Magnetic Dipoles ⃗ ⃗ ⋅ 𝑑𝑨 Φ𝐵 = ∫ ⃗𝑩 ⃗𝑭 = 𝑞𝒗 ⃗ ⃗ × ⃗𝑩 ⃗ ⃗ = 𝐼𝑨 𝝁 ⃗ =0 ⃗ ⋅ 𝑑𝑨 Φ𝐵 = ∮ ⃗𝑩 ⃗𝑭 = 𝐼 𝒍 × ⃗𝑩 ⃗ ⃗ ⃗ =𝝁 ⃗ × ⃗𝑩 𝝉 ⃗ = 𝐼𝑑𝒍 × ⃗𝑩 ⃗ 𝑑𝑭 ⃗ ⃗ ⋅ ⃗𝑩 𝑈 = −𝝁 ⃗ =𝒏 ̂ 𝑑𝐴 𝑑𝑨 Hall Effect 𝑛𝑞 = Ampere’s Law (without 𝑖𝐷 ) Source of Magnetic Fields B due to Currents ⃗⃗ ⋅ 𝑑𝒍 = 𝜇0 𝐼𝑒𝑛𝑐𝑙 ∮𝑩 ⃗⃗ = 𝜇0 𝑞𝒗⃗ ×2 𝒓̂ 𝑩 4𝜋 𝑟 𝐵= ⃗⃗ = 𝜇0 𝐼𝑑𝒍 2× 𝒓̂ 𝑑𝑩 4𝜋 𝑟 𝐵𝑥 = 2(𝑥02𝑙𝑜𝑜𝑝 +𝑎2 )3/2 𝜇0 𝐼𝑙𝑖𝑛𝑒 2𝜋𝑟 𝜇 𝐼 −𝐽𝑥 𝐵𝑦 𝐸𝑧 𝐵 = 𝜇0 𝑛𝐼𝑠𝑜𝑙. 𝑎2 𝐵= 𝜇0 𝑁𝐼𝑡𝑜𝑟. 2𝜋𝑟 ⃗ is in a magnetic material, 𝜇0 is replaced everywhere with 𝐾𝑚 𝜇0 (𝐾𝑚 is the relative permeability). When ⃗𝑩 Faraday’s Law ℰ=− 𝑑Φ𝐵 𝑑𝑡 Motional emf (General case) ∮ ⃗𝑬 ⋅ 𝑑𝒍 = − 𝑑Φ𝐵 𝑑𝑡 ℰ = 𝑣𝐵𝐿 ⃗ ) ⋅ 𝑑𝒍 (General case) ⃗ × ⃗𝑩 ℰ = ∮(𝒗 Maxwell’s Equations in a vacuum, in integral form (Click here to see the equations in differential form.) ⃗ = ⃗ ⋅ 𝑑𝑨 ∮𝑬 𝑄𝑒𝑛𝑐 𝜖0 ⃗ Gauss’s law for 𝑬 ⃗ =0 ⃗ ⋅ 𝑑𝑨 ∮ ⃗𝑩 ∮ ⃗𝑬 ⋅ 𝑑𝒍 = − ⃗ Gauss’s law for ⃗𝑩 𝑑Φ𝐵 𝑑𝑡 Faraday’s law ⃗⃗ ⋅ 𝑑𝒍 = 𝜇0 (𝑖𝐶 + 𝜖0 ∮𝑩 𝑑Φ𝐸 ) 𝑑𝑡 𝑒𝑛𝑐𝑙 where 𝑖𝐷 = 𝜖0 𝑑Φ𝐸 𝑑𝑡 Ampere’s law, including 𝑖𝐷 Chapter 15 - Waves Speed of Waves Intensity 𝑦(𝑥, 𝑡) = 𝐴 cos(𝑘𝑥 − 𝜔𝑡) 𝑠𝑝𝑒𝑒𝑑 = 𝜆𝑓 𝐼= Electromagnetic Waves Speed of EM Waves ⃗ Intensity and 𝑺 ⃗ (𝑥, 𝑡) = 𝒋̂ 𝐸𝑚𝑎𝑥 cos(𝑘𝑥 − 𝜔𝑡) 𝑬 𝑐= ̂𝐵𝑚𝑎𝑥 cos(𝑘𝑥 − 𝜔𝑡) ⃗⃗ (𝑥, 𝑡) = 𝒌 𝑩 𝑣= 𝑘= 2𝜋 𝜆 𝐸𝑚𝑎𝑥 = 𝑐𝐵𝑚𝑎𝑥 and 𝜔 = 2𝜋𝑓 1 √𝜖0 𝜇0 ≅ 3.0 × 108 𝑐 Radiation Pressure 𝐼 ⃗ = 1𝑬 ⃗ ×𝑩 ⃗⃗ 𝑺 𝜇 0 𝐼 = 𝑆𝑎𝑣 = √𝐾 𝐾𝑚 𝑝𝑟𝑎𝑑 = 𝑐 ; 𝑝𝑟𝑎𝑑 = 𝑚 𝑠 𝑃𝑎𝑣 𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑃𝑜𝑤𝑒𝑟 ( ) 𝐴 𝐴𝑟𝑒𝑎 𝐸𝑚𝑎𝑥 𝐵𝑚𝑎𝑥 2𝜇0 Momentum Flow Rate 2𝐼 𝑐 1 𝑑𝑝 𝐴 𝑑𝑡 𝑆 𝐸𝐵 0𝑐 =𝑐=𝜇 Law of Reflection Law of Refraction, Critical Angle Index of Refraction 𝜃𝑟 = 𝜃𝑎 𝑛𝑎 sin 𝜃𝑎 = 𝑛𝑏 sin 𝜃𝑏 𝑛 = 𝑣 ; 𝑛 = √𝐾𝐾𝑚 𝑛 sin 𝜃𝑐𝑟𝑖𝑡 = 𝑛𝑏 𝜆= 𝑎 Polarization 𝑐 𝐼 = 𝐼𝑚𝑎𝑥 cos 2 𝜙 𝜆0 𝑛 tan 𝜃𝑝 = 𝑛𝑏 𝑛 𝑎 Geometric Optics Equations Gaussian Sign Convention. (Don’t use the Cartesian sign convention in this course.) 𝑠 > 0 when the object is on the same side of the surface as the incoming light; 𝑠 < 0 otherwise. 𝑠′ > 0 when the image is on the same side of the surface as the outgoing light; 𝑠′ < 0 otherwise. 𝑅 > 0 when the center of curvature is on the same side as the outgoing light; 𝑅 < 0 otherwise. Gaussian Equations Spherical Mirror* Spherical Refracting Surface* 𝑛𝑎 𝑛𝑏 𝑛𝑏 − 𝑛𝑎 + ′ = 𝑠 𝑠 𝑅 ′ 𝑦 𝑛𝑎 𝑠 ′ 𝑚= =− 𝑦 𝑛𝑏 𝑠 Thin Lens 1 1 2 1 1 1 1 Object and image + ′= = + = distances 𝑠 𝑠 𝑅 𝑓 𝑠 𝑠′ 𝑓 𝑦′ 𝑠′ 𝑦′ 𝑠′ Lateral 𝑚= =− 𝑚= =− magnification 𝑦 𝑠 𝑦 𝑠 1 1 1 Lensmaker’s = (𝑛 − 1) ( − ) equation 𝑓 𝑅1 𝑅2 *The equations for a plane mirror and a plane refracting surface are obtained by setting 𝑅 = ∞. Camera Lens Simple Magnifier 𝑓-number = 𝑓/𝐷 𝑀= 𝜃′ 𝜃 = Microscope 25 𝑐𝑚 𝑓 𝑀 = 𝑚1 𝑀2 = Two-Source Interference Amplitude and Intensity 𝑚 = 0, ±1, ±2, … 𝐸𝑃 = 2𝐸 |cos | 2 𝑚𝜆 𝑑 1 𝑑 sin 𝜃 = (𝑚 + 2) 𝜆 𝐼 = 𝐼0 [ (𝑚 = ±1, ±2, ±3, … ) sin(𝛽/2) 2 ] 𝛽/2 ; 𝛽= 2𝜋 𝑎 sin 𝜃 𝜆 Circular Aperture Diffraction 𝜆 sin 𝜃1 = 1.22 𝐷 𝜙 2𝜋 (𝑟2 𝜆 𝑀=− 𝑓1 𝑓2 𝐼 = 𝐼0 cos2 2 𝜙= Single-Slit Diffraction 𝑚𝜆 𝑎 (25 𝑐𝑚) 𝑠′ 1 𝜙 𝑑 sin 𝜃 = 𝑚𝜆 ∶ 𝑦𝑚 = 𝑅 sin 𝜃 = Telescope 𝑓1 𝑓2 Thin-Film Interference 𝑚 = 1, 2, 3, … 2𝑡 = 𝑚𝜆 − 𝑟1 ) 1 2𝑡 = (𝑚 + 2) 𝜆 Multiple-Slit Diffraction X-Ray Diffraction 𝑑 sin 𝜃 = 𝑚𝜆 (𝑚 = 0, ±1, ±2, … ) 𝑚 = 1, 2, 3, … 𝜙 𝐼 = 𝐼0 cos2 2 [ sin(𝛽/2) 2 ] 𝛽/2 𝜙= 2𝜋 𝑑 sin 𝜃 𝜆 𝛽= 2𝜋 𝑎 sin 𝜃 𝜆 2𝑑 sin 𝜃 = 𝑚𝜆
