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Physics Formula Sheet for Physics For Scientists & Engineers, 4E (Knight)...

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Phy s ics Fo rm ula She et Chapter 1: Introduction: The Nature of Science and Physics

−𝑏 ± √𝑏 2 − 4𝑎𝑐 2𝑎 𝑅𝑎𝑑𝑖𝑢𝑠 𝑜𝑓 𝐸𝑎𝑟𝑡ℎ = 6.38 × 106 𝑚 𝑥=

𝑀𝑎𝑠𝑠 𝑜𝑓 𝐸𝑎𝑟𝑡ℎ = 5.98 × 1024 𝑘𝑔 𝑐 = 3.00 × 108 𝑚/𝑠 𝑁𝑚 2 𝐺 = 6.673 × 10−11 𝑘𝑔2 23 𝑁𝐴 = 6.02 × 10 𝑘 = 1.38 × 10−23 𝐽/𝐾 𝐽 𝑅 = 8.31 ⁄𝑚𝑜𝑙 ⋅ 𝐾 𝜎 = 5.67 × 10−8 𝑊/(𝑚2 ⋅ 𝐾) 𝑘 = 8.99 × 10 𝑁 ⋅ 𝑚 /𝐶 𝑞𝑒 = −1.60 × 10−19 𝐶 𝜖0 = 8.85 × 10−12 𝐶 2 /(𝑁 ⋅ 𝑚 2 ) 𝜇0 = 4π × 10−7 𝑇 ⋅ 𝑚/𝐴 ℎ = 6.63 × 10−34 𝐽 ⋅ 𝑠 𝑚𝑒 = 9.11 × 10−31 𝑘𝑔 𝑚𝑝 = 1.6726 × 10−27 𝑘𝑔 9

2

2

𝑚𝑛 = 1.6749 × 10−27 𝑘𝑔

𝑎𝑚𝑢 = 1.6605 × 10−27 𝑘𝑔 𝑘𝑔 𝐷𝑒𝑛𝑠𝑖𝑡𝑦 𝑜𝑓 𝑤𝑎𝑡𝑒𝑟 = 1000 3 𝑚

Chapter 2: Kinematics 𝛥𝑥 = 𝑥𝑓 − 𝑥0

𝛥𝑡 = 𝑡𝑓 − 𝑡0 𝛥𝑥 𝑥𝑓 − 𝑥0 𝑣= = 𝑡𝑓 − 𝑡0 𝛥𝑡 𝛥𝑣 𝑣𝑓 − 𝑣0 𝑎= = 𝑡𝑓 − 𝑡0 𝛥𝑡

𝑥 = 𝑥0 + 𝑣 𝑡 𝑣0 + 𝑣 𝑣= 2 𝑣 = 𝑣0 + 𝑎𝑡 1 𝑥 = 𝑥0 + 𝑣0 𝑡 + 𝑎𝑡 2 2 𝑣 2 = 𝑣02 + 2𝑎 (𝑥 − 𝑥0 ) 𝑚 𝑔 = 9.80 2 𝑠

Chapter 3: Two-Dimensional Kinematics 𝐴𝑥 = 𝐴 𝑐𝑜𝑠 𝜃 𝐴𝑦 = 𝐴 𝑠𝑖𝑛 𝜃

𝑅𝑥 = 𝐴𝑥 + 𝐵𝑥

𝑅𝑦 = 𝐴𝑦 + 𝐵𝑦

𝑅 = √𝑅2𝑥 + 𝑅𝑦2 𝜃 = 𝑡𝑎𝑛−1

𝑅𝑦 𝑅𝑥

2 𝑣0𝑦 2𝑔 2 𝑣0 𝑠𝑖𝑛 2𝜃0 𝑅= 𝑔 𝑣𝑥 = 𝑣 𝑐𝑜𝑠 𝜃 𝑣𝑦 = 𝑣 𝑠𝑖𝑛 𝜃

ℎ=

𝑣 = √𝑣𝑥2 + 𝑣𝑦2 𝜃 = 𝑡𝑎𝑛−1

𝑣𝑦 𝑣𝑥

Chapter 4: Dynamics: Forces and Newton’s Laws of Motion 𝐹𝑛𝑒𝑡 = 𝑚𝑎 𝑤 = 𝑚𝑔

Chapter 5: Further Applications of Newton’s Laws: Friction, Drag, and Elasticity 𝑓𝑠 ≤ 𝜇𝑠 𝑁 𝑓𝑘 = 𝜇𝑘 𝑁 1 𝐹𝐷 = 𝐶𝜌𝐴𝑣 2 2 𝐹𝑠 = 6𝜋𝜂𝑟𝑣 𝐹 = 𝑘𝛥𝑥 1𝐹 𝛥𝐿 = 𝐿 𝑌𝐴 0 𝐹 𝑠𝑡𝑟𝑒𝑠𝑠 = 𝐴 𝛥𝐿 𝑠𝑡𝑟𝑎𝑖𝑛 = 𝐿0 𝑠𝑡𝑟𝑒𝑠𝑠 = 𝑌 × 𝑠𝑡𝑟𝑎𝑖𝑛 1𝐹 𝐿 𝛥𝑥 = 𝑆𝐴 0 1𝐹 𝑉 𝛥𝑉 = 𝐵𝐴 0

Chapter 6: Uniform Circular Motion and Gravitation 𝛥𝜃 =

𝛥𝑠

𝑟 2𝜋 𝑟𝑎𝑑 = 360° = 1 𝑟𝑒𝑣𝑜𝑙𝑢𝑡𝑖𝑜𝑛 𝛥𝜃 𝜔= 𝛥𝑡

𝑣 = 𝑟𝜔

𝑣2 𝑟 𝑎𝐶 = 𝑟𝜔2 𝐹𝐶 = 𝑚𝑎𝐶 𝑚𝑣 2 𝐹𝐶 = 𝑟 𝑣2 𝑡𝑎𝑛 𝜃 = 𝑟𝑔 𝑎𝐶 =

𝐹𝐶 = 𝑚𝑟𝜔2 𝑚𝑀 𝐹=𝐺 2 𝑟 𝐺𝑀 𝑔= 2 𝑟 𝑇12 𝑟13 = 𝑇22 𝑟23 4𝜋 2 3 𝑇2 = 𝑟 𝐺𝑀 𝑟3 𝐺 = 𝑀 2 𝑇 4𝜋 2

Chapter 7: Work, Energy, and Energy Resources 𝑊 = 𝑓𝑑 𝑐𝑜𝑠 𝜃 1 𝐾𝐸 = 𝑚𝑣 2 2 1 1 𝑊𝑛𝑒𝑡 = 𝑚𝑣𝑓2 − 𝑚𝑣02 2 2 𝑃𝐸𝑔 = 𝑚𝑔ℎ 1 𝑃𝐸𝑠 = 𝑘𝑥 2 2 𝐾𝐸0 + 𝑃𝐸0 = 𝐾𝐸𝑓 + 𝑃𝐸𝑓

𝐾𝐸0 + 𝑃𝐸0 + 𝑊𝑛𝑐 = 𝐾𝐸𝑓 + 𝑃𝐸𝑓 𝑊𝑜𝑢𝑡 𝐸𝑓𝑓 = 𝐸𝑖𝑛 𝑊 𝑃= 𝑡

Chapter 8: Linear Momentum and Collisions 𝑝 = 𝑚𝑣 𝛥𝑝 = 𝐹𝑛𝑒𝑡 𝛥𝑡 𝑝0 = 𝑝𝑓

𝑚1 𝑣01 + 𝑚2 𝑣02 = 𝑚1 𝑣𝑓1 + 𝑚2 𝑣𝑓2

2 1 1 2 𝑚1 𝑣01 + 𝑚2 𝑣02 2 1 2 2 = 𝑚1 𝑣𝑓1 2 2 1 + 𝑚2 𝑣𝑓2 2 𝑚1 𝑣1 = 𝑚1 𝑣1′ 𝑐𝑜𝑠 𝜃1 + 𝑚2 𝑣2′ 𝑐𝑜𝑠 𝜃2 0 = 𝑚1 𝑣1′ 𝑠𝑖𝑛 𝜃1 + 𝑚2 𝑣2′ 𝑠𝑖𝑛 𝜃2 1 1 1 𝑚𝑣12 = 𝑚𝑣1′2 + 𝑚𝑣2′2 2 2 2 + 𝑚𝑣1′ 𝑣2′ 𝑐𝑜𝑠(𝜃1 − 𝜃2 ) 𝑣𝑒 𝛥𝑚 𝑎= −𝑔 𝑚 𝛥𝑡 𝑣1 𝑚1 + 𝑣2 𝑚2 𝑣𝑐𝑚 = 𝑚1 + 𝑚2

Chapter 9: Statics and Torque 𝜏 = 𝑟𝐹 𝑠𝑖𝑛 𝜃 𝑟⊥ = 𝑟 𝑠𝑖𝑛 𝜃 𝐹𝑜 𝑙𝑖 𝑀𝐴 = = 𝐹𝑖 𝑙𝑜 𝑙𝑖 𝐹𝑖 = 𝑙𝑜 𝐹𝑜 𝛥𝜃

(𝑅12 + 𝑅22 )

cylinder axis: 𝐼 =

𝑀𝑅 2 2

Solid cylinder (or disk) about

central diameter: 𝐼 =

𝑀ℓ2

𝑀ℓ2

3 2𝑀𝑅 2 5

Thin spherical shell: 𝐼 =

2𝑀𝑅 2 3

Slab about ⊥ axis through center: 𝐼=

𝑀(𝑎2 +𝑏 2) 12

𝑛𝑒𝑡 𝑊 = (𝑛𝑒𝑡 𝜏)𝜃 1 𝐾𝐸𝑟𝑜𝑡 = 𝐼𝜔2 2 𝐿 = 𝐼𝜔 𝛥𝐿 𝑛𝑒𝑡 𝜏 = 𝛥𝑡 𝑚 𝑉 𝐹

𝐴 𝑃𝑎𝑡𝑚 = 1.01 × 105 𝑃𝑎 𝑃 = 𝜌𝑔ℎ 𝑃2 = 𝑃1 + 𝜌𝑔ℎ 𝐹1 𝐹2 = 𝐴1 𝐴2 𝐹𝐵 = 𝑤𝑓𝑙 𝜌𝑜𝑏𝑗 𝐹𝑟𝑎𝑐𝑡𝑖𝑜𝑛 𝑠𝑢𝑏𝑚𝑒𝑟𝑔𝑒𝑑 = 𝜌𝑓𝑙 𝑠𝑝𝑒𝑐𝑖𝑓𝑖𝑐 𝑔𝑟𝑎𝑣𝑖𝑡𝑦 =

Hoop about any diameter: 𝐼 = 2

Solid sphere: 𝐼 =

𝑃=

𝜔 = 𝜔0 + 𝛼𝑡 1 𝜃 = 𝜔0 𝑡 + 𝛼𝑡 2 2 𝜔2 = 𝜔02 + 2𝛼𝜃 𝜔0 + 𝜔 𝜔= 2 𝑛𝑒𝑡 𝜏 = 𝐼𝛼 Hoop about cylinder axis: 𝐼 = 𝑀𝑅2 𝑀

⊥ to length: 𝐼 =

𝜌=

𝛥𝑡 𝑣 = 𝑟𝜔 𝛥𝜔 𝛼= 𝛥𝑡 𝛥𝑣 𝑎𝑡 = 𝛥𝑡 𝑎𝑡 = 𝑟𝛼 𝜃 = 𝜔𝑡

Ring: 𝐼 =

⊥ to length: 𝐼 = 12 Thin rod about axis through one end

Chapter 11: Fluid Statics

Chapter 10: Rotational Motion and Angular Momentum 𝜔=

Please Do Not Write on This Sheet Thin rod about axis through center

𝑀𝑅 2 2

+

Solid cylinder (or disk) about 𝑀𝑅2 4

𝑀ℓ2 12

𝛾=

𝑃=

ℎ=

𝐹 𝐿 4𝛾

𝜌 𝜌𝑤

𝑟 2𝛾 𝑐𝑜𝑠 𝜃 𝜌𝑔𝑟

Chapter 12: Fluid Dynamics and Its Biological Medical Applications 𝑉 𝑄= 𝑡 𝑄 = 𝐴𝑣 𝐴1 𝑣1 = 𝐴2 𝑣2 𝑛1 𝐴1 𝑣1 = 𝑛2 𝐴2 𝑣2

1 𝑃1 + 𝜌𝑣12 + 𝜌𝑔ℎ1 2 1 = 𝑃2 + 𝜌𝑣22 2 + 𝜌𝑔ℎ2 1 2 (Δ𝑃 + Δ 𝜌𝑣 + Δ𝜌𝑔ℎ) 𝑄 = 𝑝𝑜𝑤𝑒𝑟 2 𝑣1 = √2𝑔ℎ 𝐹𝐿 𝜂= 𝑣𝐴 𝑃2 − 𝑃1 𝑄= 𝑅 8𝜂𝑙 𝑅= 4 𝜋𝑟 (𝑃2 − 𝑃1 )𝜋𝑟 4 𝑄= 8𝜂𝑙 2𝜌𝑣𝑟 𝑁𝑅 = 𝜂 𝜌𝑣𝐿 𝑁𝑅′ = 𝜂 𝑥𝑟𝑚𝑠 = √2𝐷𝑡

Chapter 13: Temperature, Kinetic Theory, and the Gas Laws 9 𝑇(°𝐶) + 32 5 𝑇(𝐾) = 𝑇 (°𝐶 ) + 273.15 𝛥𝐿 = 𝛼𝐿𝛥𝑇 𝛥𝐴 = 2𝛼𝐴𝛥𝑇 𝛥𝑉 = 𝛽𝑉𝛥𝑇 𝛽 ≈ 3𝛼 𝑃𝑉 = 𝑁𝑘𝑇 𝑘 = 1.38 × 10−23 𝐽/𝐾 𝑁𝐴 = 6.02 × 1023 𝑚𝑜𝑙 −1 𝑃𝑉 = 𝑛𝑅𝑇 𝐽 𝑅 = 8.31 𝑚𝑜𝑙 ⋅ 𝐾 1 2 𝑃𝑉 = 𝑁𝑚𝑣 3 3 1 2 𝐾𝐸 = 𝑚𝑣 = 𝑘𝑇 2 2 𝑇(°𝐹) =

𝑣𝑟𝑚𝑠 = √

3𝑘𝑇 𝑚

% 𝑟𝑒𝑙𝑎𝑡𝑖𝑣𝑒 ℎ𝑢𝑚𝑖𝑑𝑖𝑡𝑦 𝑣𝑎𝑝𝑜𝑟 𝑑𝑒𝑛𝑠𝑖𝑡𝑦 = 𝑠𝑎𝑡𝑢𝑟𝑎𝑡𝑖𝑜𝑛 𝑣𝑎𝑝𝑜𝑟 𝑑𝑒𝑛𝑎𝑠𝑖𝑡𝑦 × 100%

Chapter 14: Heat and Heat Transfer Methods

1.000 𝑘𝑐𝑎𝑙 = 4186 𝐽 𝑄 = 𝑚𝑐𝛥𝑇 𝑄 = 𝑚𝐿𝑓 𝑄 = 𝑚𝐿𝑣 𝑄 𝑘𝐴(𝑇2 − 𝑇1 ) = 𝑑 𝑡 𝑄 = 𝜎𝑒𝐴𝑇 4 𝑡 𝐽 𝜎 = 5.67 × 10−8 𝑠 ⋅ 𝑚2 ⋅ 𝐾 4 𝑄𝑛𝑒𝑡 = 𝜎𝑒𝐴(𝑇24 − 𝑇14 ) 𝑡

Chapter 15: Thermodynamics

3 𝑈 = 𝑁𝑘𝑇 2 𝛥𝑈 = 𝑄 − 𝑊 𝑊 = 𝑃𝛥𝑉 (𝑖𝑠𝑜𝑏𝑎𝑟𝑖𝑐 𝑝𝑟𝑜𝑐𝑒𝑠𝑠) Δ𝑈 = 𝑄 − 𝑃Δ𝑉 𝑊 = 0 (𝑖𝑠𝑜𝑐ℎ𝑜𝑟𝑖𝑐 𝑝𝑟𝑜𝑐𝑒𝑠𝑠) Δ𝑈 = 𝑄 𝑄 = 𝑊 (𝑖𝑠𝑜𝑡ℎ𝑒𝑟𝑚𝑎𝑙 𝑝𝑟𝑜𝑐𝑒𝑠𝑠) Δ𝑈 = 0 ( 𝑄 = 0 𝑎𝑑𝑖𝑎𝑏𝑎𝑡𝑖𝑐 𝑝𝑟𝑜𝑐𝑒𝑠𝑠) Δ𝑈 = −𝑊 𝑊 𝐸𝑓𝑓 = 𝑄ℎ 𝑄𝑐 (𝑐𝑦𝑐𝑙𝑖𝑐𝑎𝑙 𝑝𝑟𝑜𝑐𝑒𝑠𝑠) 𝐸𝑓𝑓 = 1 − 𝑄ℎ 𝑇𝑐 𝐸𝑓𝑓𝐶 = 1 − 𝑇ℎ 𝑄ℎ 𝐶𝑂𝑃ℎ𝑝 = 𝑊 𝑄𝑐 𝐶𝑂𝑃𝑟𝑒𝑓 = 𝐶𝑂𝑃ℎ𝑝 − 1 = 𝑊 𝑄𝑐⁄ 𝑡1 𝐸𝐸𝑅 = 𝑄ℎ ⁄𝑡2 𝑄 𝛥𝑆 = 𝑇 𝑄ℎ 𝑄𝑐 𝛥𝑆𝑡𝑜𝑡 = + =0 𝑇ℎ 𝑇𝑐 𝑊𝑢𝑛𝑎𝑣𝑎𝑖𝑙 = 𝛥𝑆 ⋅ 𝑇0 𝑆 = 𝑘 𝑙𝑛 𝑊 𝑘 = 1.38 × 10−23 𝐽/𝐾

Chapter 16: Oscillatory Motion and Waves 1 𝑓= 𝑇 𝜆 𝑣 = = 𝑓𝜆 𝑇 𝐹 = −𝑘𝑥

Please Do Not Write on This Sheet

𝑃𝐸 𝑉= 𝑞 𝛥𝑃𝐸 = 𝑞𝛥𝑉 𝑊 = 𝑞𝑉𝐴𝐵

1 𝑃𝐸𝑒𝑙 = 𝑘𝑥 2 2 𝑚 𝑇 = 2𝜋√ 𝑘 𝑓=

1

2𝜋

𝑉𝐴𝐵 𝐸= 𝑑 𝛥𝑉 𝐸=− 𝛥𝑠 𝑘𝑄 𝑉= 𝑟 𝑄 𝐶= 𝑉 𝐴 𝐶 = 𝜖0 𝑑

𝑘 √𝑚

2𝜋𝑡 ) 𝑇 2𝜋𝑡 𝑣(𝑡) = −𝑣𝑚𝑎𝑥 𝑠𝑖𝑛 ( ) 𝑇 𝑥(𝑡) = 𝑋 𝑐𝑜𝑠 (

𝑣𝑚𝑎𝑥 =

2𝜋𝑋 𝑘 = 𝑋√ 𝑇 𝑚

𝑎(𝑡) = −

𝑘𝑋 2𝜋𝑡 ) 𝑐𝑜𝑠 ( 𝑇 𝑚

𝑣𝑠𝑡𝑟𝑖𝑛𝑔 = √

𝐹

𝑚/𝐿

𝑚 𝑇 𝑣𝑤 = (331 ) √ 273 𝐾 𝑠 𝐼=

𝑃

𝐴 𝐴𝑠𝑝ℎ𝑒𝑟𝑒 = 4𝜋𝑟 2 (𝛥𝑝)2 𝐼= 2𝜌𝑣𝑤

Chapter 17: Physics of Hearing 𝐼 𝛽 = (10 𝑑𝐵) 𝑙𝑜𝑔 ( ) 𝐼0 𝑣𝑤 ± 𝑣𝑜 ) 𝑓𝑜 = 𝑓𝑠 ( 𝑣𝑤 ∓ 𝑣𝑠 𝑓𝐵 = |𝑓1 − 𝑓2 | 𝑣𝑤 𝑓𝑛 = 𝑛 ( ) 2𝐿 𝑣𝑤 𝑓𝑛 = 𝑛 ( ) 4𝐿 𝑍 = 𝜌𝑣 (𝑍2 − 𝑍1 )2 𝑎= (𝑍1 + 𝑍2 )2

Chapter 18: Electric Charge and Electric Field |𝑞𝑒 | = 1.60 × 10−19 𝐶 |𝑞1 𝑞2 | 𝐹=𝑘 2 𝑟 𝐸 = 𝐹/𝑞 |𝑄| 𝐸=𝑘 2 𝑟

Chapter 19: Electric Potential and Electric Energy

𝜖0 = 8.85 × 10−12 𝐸𝑐𝑎𝑝

𝐶 = 𝜅𝜖0

𝐴

𝐹

𝑚

𝑑 𝑄𝑉 𝐶𝑉 2 𝑄2 = = = 2 2 2𝐶

Chapter 20: Electric Current, Resistance, and Ohm’s Law 𝐼=

𝛥𝑄

𝛥𝑡 𝐼 = 𝑛𝑞𝐴𝑣𝑑 𝑉 = 𝐼𝑅 𝜌𝐿 𝑅= 𝐴 𝜌 = 𝜌0 (1 + 𝛼𝛥𝑇) 𝑅 = 𝑅0 (1 + 𝛼𝛥𝑇) 𝑉2 = 𝐼2 𝑅 𝑃 = 𝐼𝑉 = 𝑅 1 𝑃𝑎𝑣𝑒 = 𝐼0 𝑉0 2 𝐼0 𝐼𝑟𝑚𝑠 = √2 𝑉0 𝑉𝑟𝑚𝑠 = √2

Chapter 21: Circuits, Bioelectricity, and DC Instruments

𝑅𝑆 = 𝑅1 + 𝑅2 + 𝑅3 + ⋯ 1 1 1 1 + +⋯ + = 𝑅 𝑅𝑃 𝑅1 𝑅2 3 𝑉 = 𝑒𝑚𝑓 − 𝐼𝑟 𝑡

𝑉 = 𝑒𝑚𝑓 (1 − 𝑒 − 𝑅𝐶 ) 𝜏 = 𝑅𝐶

𝑡

𝑉 = 𝑉0 𝑒 − 𝑟𝐶

Please Do Not Write on This Sheet

Chapter 22: Magnetism 𝐹 = 𝑞𝑣𝐵 𝑠𝑖𝑛 𝜃 𝑚𝑣 𝑟 = 𝑞𝐵 𝜖 = 𝐵𝑙𝑣 𝐹 = 𝐼𝐿𝐵 𝑠𝑖𝑛 𝜃 𝜏 = 𝑁𝐼𝐴𝐵 𝑠𝑖𝑛 𝜃 𝜇0 𝐼 𝐵= 2𝜋𝑟 𝜇0 𝐼 𝐵= 2𝑅 𝐵 = 𝜇0 𝑛𝐼 𝐹 𝜇0 𝐼1 𝐼2 = 𝑙 2𝜋𝑟

Chapter 23: Electromagnetic Induction, AC Circuits, and Electrical Technologies 𝛷 = 𝐵𝐴 𝑐𝑜𝑠 𝜃 𝛥𝛷 𝑒𝑚𝑓 = −𝑁 𝛥𝑡 𝑒𝑚𝑓 = 𝑣𝐵𝐿 𝑒𝑚𝑓 = 𝑁𝐴𝐵𝜔 𝑠𝑖𝑛 𝜔𝑡 𝑉𝑆 𝐼𝑃 𝑁𝑆 = = 𝑉𝑃 𝑁𝑃 𝐼𝑆 𝛥𝐼2 𝑒𝑚𝑓1 = −𝑀 𝛥𝑡 𝛥𝐼 𝑒𝑚𝑓 = −𝐿 𝛥𝑡 𝛥𝛷 𝐿=𝑁 𝛥𝐼 μ0 𝑁 2 𝐴 𝐿= ℓ 1 2 𝐸𝑖𝑛𝑑 = 𝐿𝐼 2 𝑡

𝐼 = 𝐼0 (1 − 𝑒 −𝜏 ) 𝜏=

𝐿 𝑅

𝑡

𝐼 = 𝐼0 𝑒 −𝜏 𝑉 𝐼= 𝑋𝐿 𝑋𝐿 = 2𝜋𝑓𝐿 𝑉 𝐼= 𝑋𝐶 1 𝑋𝐶 = 2𝜋𝑓𝐶 𝑉0 𝑉𝑟𝑚𝑠 𝐼0 = 𝑜𝑟 𝐼𝑟𝑚𝑠 = 𝑍 𝑍 𝑍 = √𝑅2 + (𝑋𝐿 − 𝑋𝐶 )2 1 𝑓0 = 2𝜋√𝐿𝐶

𝑅 𝑐𝑜𝑠 𝜙 = 𝑍 𝑃𝑎𝑣𝑒 = 𝐼𝑟𝑚𝑠 𝑉𝑟𝑚𝑠 𝑐𝑜𝑠 𝜙

Chapter 24: Electromagnetic Waves 𝑐=

1

√𝜇0 𝜖0 𝐸 =𝑐 𝐵 𝑐 = 𝑓𝜆 𝑐𝜖0 𝐸02 𝐼𝑎𝑣𝑒 = 2 𝑐𝐵02 𝐼𝑎𝑣𝑒 = 2𝜇0 𝐸0 𝐵0 𝐼𝑎𝑏𝑒 = 2𝜇0

Chapter 25: Geometric Optics 𝜃𝑖 = 𝜃𝑟 𝑐 𝑛= 𝑣 𝑛1 𝑠𝑖𝑛 𝜃1 = 𝑛2 𝑠𝑖𝑛 𝜃2 𝑛2 𝜃𝑐 = 𝑠𝑖𝑛−1 𝑛1 1 𝑃= 𝑓 1 1 1 + = 𝑓 𝑑𝑜 𝑑𝑖 ℎ𝑖 𝑑𝑖 𝑚= =− ℎ𝑜 𝑑𝑜 𝑅 𝑓= 2

Chapter 26: Vision and Optical Instruments 1 1 + 𝑑𝑜 𝑑𝑖 𝑚 = 𝑚𝑜 𝑚 𝑒 𝑁𝐴 = 𝑛 𝑠𝑖𝑛 𝛼 𝑓 1 𝑓/# = ≈ 𝐷 2𝑁𝐴 𝑑𝑖 = 𝑓𝑜 𝑓𝑜 𝑀= 𝑓𝑒 𝑃=

Chapter 27: Wave Optics 𝜆𝑛 =

𝜆 𝑛

sin 𝜃 = 𝑚

𝜆 𝑑

1 𝜆 𝑠𝑖𝑛 𝜃 = (𝑚 + ) 2 𝜆 𝑑 𝑠𝑖𝑛 𝜃 = 𝑚 𝑊 𝜆 𝜃 = 1.22 𝐷 𝜆𝑛 2𝑡 = 2 2𝑡 = 𝜆𝑛 I = ½ I0 𝐼 = 𝐼0 𝑐𝑜𝑠 2 𝜃 𝑛2 𝑡𝑎𝑛 𝜃𝑏 = 𝑛1

Chapter 28: Special Relativity 𝛥𝑡 = 𝛾=

𝛥𝑡0

√1 − 𝑣 2 𝑐 1

√1 − 𝑣 2 𝑐

𝐿 = 𝐿0 √1 −

2

𝑣2 𝑐2

𝑣𝐿𝑇 + 𝑣 𝑇𝐺 𝑣 𝑣 1 + 𝐿𝑇 2 𝑇𝐺 𝑐 𝑢 1+ 𝑐 = 𝜆𝑠 √ 𝑢 1− 𝑐

𝑣𝐿𝐺 = 𝜆𝑜𝑏𝑠

2

𝑢 1− 𝑐 𝑓𝑜𝑏𝑠 = 𝑓𝑠 √ 𝑢 1+ 𝑐 𝑚𝑣 𝑝= 2 √1 − 𝑣 2 𝑐 𝑚𝑐 2 𝐸= 2 √1 − 𝑣 2 𝑐 𝐸0 = 𝑚𝑐 2 𝑚𝑐2 𝐾𝐸𝑟𝑒𝑙 = − 𝑚𝑐 2 2 𝑣 √1 − 2 𝑐 𝐸 2 = (𝑝𝑐)2 + (𝑚𝑐 2 )2...