2.3.32
Intuition The Big Picture (YE KYUN EXIST KARTA HAI)Newton kehta tha ki energy aur momentum alag-alag, simple cheezein hain (K E = 2 1 m v 2 p = m v ek hi object ke do chehere hain — energy-momentum 4-vector. Jaise ek stick ki length tab nahi badlti jab tum use rotate karo, waise hi energy-momentum vector ki "length" invariant hoti hai : ye hamesha m c 2 hi rest mass hai. Poora formula E 2 = ( p c ) 2 + ( m c 2 ) 2 is 4-vector ke liye Pythagorean theorem hai .
E total relativistic energy (rest + kinetic). Units: joules.p p = γ m v m rest mass (invariant mass — har frame mein same).c m c 2 rest energy .γ = 1 − v 2 / c 2 1
Intuition Ye Pythagorean theorem kyun haiEk right triangle socho: hypotenuse E p c m c 2 m c 2 fixed hai (rest energy), leg p c E
Worked example Example 1 — Electron ki rest energym = 9.11 × 1 0 − 31 E = m c 2 = 9.11 × 1 0 − 31 × ( 3 × 1 0 8 ) 2 = 8.2 × 1 0 − 14 J ÷ 1.6 × 1 0 − 19 = 0.511 Ye step kyun? Electron charge se divide karne par joules, electron-volts mein convert hote hain, jo particle physics mein natural unit hai.
Worked example Example 2 — Photon momentumEk photon ki energy E = 2.0 p m = 0 p = E / c p = 3 × 1 0 8 2.0 × 1.6 × 1 0 − 19 = 1.07 × 1 0 − 27 kg⋅m/s E = p c γ v c
Worked example Example 3 — Momentum aur mass se total energyEk proton (m c 2 = 938 p c = 500 E K E E = ( 500 ) 2 + ( 938 ) 2 = 250000 + 880000 ≈ 1063 MeV K E = E − m c 2 = 1063 − 938 = 125 m c 2 bachta hai.
Common mistake "Mass speed ke saath badhti hai, isliye formula meinm r e l = γ m Ye sahi kyun lagta hai: Purane textbooks ne "relativistic mass" γ m E = m r e l c 2 Fix yeh hai: E 2 = ( p c ) 2 + ( m c 2 ) 2 m invariant rest mass hai — ye kabhi nahi badlta. Agar tum yahan γ m γ
E = m c 2 Ye sahi kyun lagta hai: Ye sabse famous equation hai, toh lagta hai ye universal hogi.
Fix yeh hai: E = m c 2 rest energy hai (jab p = 0 E = ( p c ) 2 + ( m c 2 ) 2 E = γ m c 2
Common mistake "Photons mein momentum nahi hota kyunkip = m v m = 0 Ye sahi kyun lagta hai: Newtonian p = m v Fix yeh hai: Newtonian p = m v p = E / c sach mein push karti hai (radiation pressure, solar sails).
Recall Feynman: 12-saal ke bachche ko samjhaoSocho energy ek ladder ki tarah hai jo diwar se lagi hai. Diwar ki height "rest energy" hai — woh energy jo kisi cheez mein sirf exist karne ki wajah se hoti hai, chahe woh bilkul still baithi ho. Ladder ka paira kitna bahar kheencha hai woh "motion energy" (momentum) hai. Ladder ki length total energy hai. Agar paira aur bahar kheecho (zyada speed), ladder lambi ho jaati hai (zyada energy), lekin diwar ki height kabhi nahi badlti — woh unchanging mass hai. Light ek aisi ladder hai jisme koi diwar hi nahi: pure motion energy, weightless, lekin phir bhi tiki rehti hai aur push karti hai!
Mnemonic Structure yaad rakho"Energy is the Hypotenuse" — ek right triangle banao jisme E m c 2 p c E 2 = ( p c ) 2 + ( m c 2 ) 2
Worked example Compute karne se pehle predict karoQ: Jab particle ki speed → c E Forecast: γ → ∞ E → ∞ c Verify: E = γ m c 2 γ = 1/ 1 − v 2 / c 2 → ∞ v → c m c 2 = 0 exactly c
Poora energy-momentum relation kya hai? E 2 = ( p c ) 2 + ( m c 2 ) 2
E 2 = ( p c ) 2 + ( m c 2 ) 2 m Invariant rest mass (har frame mein same)
Rest par particle ke liye formula kya ban jaata hai? E = m c 2
Massless particle (photon) ke liye kya ban jaata hai? E = p c
E 2 − ( p c ) 2 Ye m 2 c 4
Formula Pythagoras jaisa kyun hai? E p c m c 2
Photon ka momentum uski energy ke terms mein express karo. p = E / c
Massive particle speed c v → c γ → ∞
Total energy se kinetic energy kaise nikalte hain? K E = E − m c 2 = ( γ − 1 ) m c 2
Relativistic momentum ki definition? Right triangle: E hypotenuse
