2.3.32 · D1Mass-energy equivalence E² = (pc)² + (mc²)²
Ek chalti hui cheez energy ko do independent tareekon se ek saath carry karti hai: sirf exist karne ki energy (uski rest energy) aur move karne ki energy (uska momentum). Ye dono ek right triangle ki do legs ki tarah combine hote hain, aur hypotenuse total energy hai — yahi E 2 = ( p c ) 2 + ( m c 2 ) 2
Is page pe koi assumption nahi hai. Agar parent note (topic note) mein koi symbol use hua tha, toh hum use yahan se ground up se build karte hain.
Kisi bhi physics se pehle, geometry ka ek piece poore topic ko power karta hai.
Definition Right triangleRight triangle wo triangle hota hai jisme ek square corner (9 0 ∘ legs hain. Corner ke opposite wali lambi tedhi side hypotenuse hai.
Do arrows imagine karo jo ek hi point se start ho rahe hain: ek seedha upar, ek seedha daayein. Ye perpendicular hain (9 0 ∘
Intuition "Perpendicular" se kya faydaDo energies squares mein add hoti hain (directly add nahi) kyunki wo perpendicular hain — wo alag directions mein point karti hain, toh ek ka badhna doosri ko nahi ghataata. Rest-energy leg fixed rehti hai jabki motion leg badhti hai. Isliye m c 2 E
v v v
c c 3 × 1 0 8 speed limit ki tarah imagine karo — ek wall jise koi bhi massive object cross nahi kar sakta. Har relativity formula v c Topic ko ye kyun chahiye: c conversion factor hai. Ye ek mass (kilograms) ko m c 2
m m jab wo still baitha ho tab measure ki gayi . Ise scale ki reading ki tarah imagine karo jab object aapke paas rest mein ho. Hum ise rest mass ya invariant mass kehte hain kyunki har koi, chahe wo kaise bhi move kar raha ho, is number par agree karta hai.
"Rest" kyun matter karta hai: puraani books mein kaha jaata tha ki mass speed ke saath "badh" jaata hai — yah topic jaanboojhkar us se bachta hai. Yahan m
p p = m v quantity of motion .
Har relativistic effect is par depend karta hai ki aapki speed light speed ke kitna close hai , yaani fraction v / c
v / c v / c 0 1 v / c = 0 v / c = 1
Ab ek tool enter karta hai. Hume naya symbol chahiye hi kyun? Kyunki high speed par, time, length, energy aur momentum sab ek hi factor se stretch hote hain, aur baar baar 1 − v 2 / c 2 1 γ
γ γ = 1 − v 2 / c 2 1 stretch dial ki tarah padho. v / c γ ≥ 1
Is dial ko har case mein padho — reader ko koi bhi aisi value nahi milni chahiye jo humne pehle na dikhaayi ho:
γ Kyun sahi lagta hai: stretching sunke lagta hai ki shrink bhi ho sakta hai.
Fix: v 2 / c 2 ≥ 0 1 − v 2 / c 2 ≤ 1 ≥ 1 γ 1 neeche nahi jaata.
Ab hum wo dono quantities assemble karte hain jo triangle ko chahiye.
E E saari energy hai jo ek object ke paas hai: exist karne ki energy plus move karne ki energy, ek hi number mein. Units: joules (J). Relativity likhta hai
E = γ m c 2 . m c 2 γ γ = 1 m c 2
m c 2 fixed vertical leg ki tarah imagine karo — ye kabhi nahi badlti kyunki m c
p c p c energy units mein (joules), toh ye E m c 2 badhti hui horizontal leg ki tarah imagine karo: rest par p = 0 c c
Yahan wo master picture hai jiske liye ye saare symbols banaye gaye the:
Intuition Sab kuch ek saath
Vertical leg m c 2 fixed (rest energy).
Horizontal leg p c badhti hai (motion energy).
Hypotenuse E ( p c ) 2 + ( m c 2 ) 2 E m c 2
Worked examples do energy units ke beech switch karte hain, toh dono abhi build karo.
Energy ki everyday scientific unit. Ek chhota sa apple ek metre upar uthane mein lagbhag ek joule lagta hai.
Definition electron-volt (eV)Particles ke liye natural ek bahut chhoti energy unit: 1 eV = 1.6 × 1 0 − 19 = 0.511 = 938 J → eV convert karo: 1.6 × 1 0 − 19 eV → J convert karo: 1.6 × 1 0 − 19 1 MeV = 1 0 6
total energy E = gamma m c squared
Pythagoras h squared = a squared + b squared
E squared = pc squared + mc squared squared
Khud test karo — answer karne ke baad hi reveal karo.
Energy triangle ki do legs aur hypotenuse kya hain? legs p c m c 2 E
Pythagoras' theorem state karo. legs a , b h h 2 = a 2 + b 2
c speed of light, cosmic speed limit, ≈ 3 × 1 0 8
m ye rest par measure kiya jaata hai aur har observer iss par agree karta hai; ye speed ke saath kabhi nahi badlta
Lorentz factor γ v = 0 γ v → c rest par γ = 1 v → c γ → ∞
γ v 2 / c 2 ≥ 0 1 − v 2 / c 2 ≤ 1 ≥ 1
Identity γ 2 = c 2 / ( c 2 − v 2 ) γ 2 = 1/ ( 1 − v 2 / c 2 ) c 2
Momentum p c kg·m/s ko energy units (joules) mein convert karne ke liye taaki ye triangle par fit ho sake
Total energy do equivalent tareekon mein likho. E = γ m c 2 aur
E = ( p c ) 2 + ( m c 2 ) 2 Ek electron-volt mein kitne joules hote hain? 1 eV = 1.6 × 1 0 − 19
Joules mein energy ko eV mein convert karo. 1.6 × 1 0 − 19
Special Relativity — jahan se c Lorentz Factor — stretch dial γ Photon Momentum — m = 0 Relativistic Kinetic Energy — bacha hua ( γ − 1 ) m c 2 Four-Momentum — wo deeper object jiska "length" m c 2 Nuclear Binding Energy — jahan mass ka energy mein convert hona real consequences rakhta hai 
