Speed of light c = 1 - √(ε₀ μ₀): Visual walkthrough

1.8.34 · D2 · Physics › Electromagnetism › Speed of light c = 1 - √(ε₀ μ₀)

Hum Maxwell's Equations ka sahara lenge, khaaskar Faraday's Law of Induction, Ampère–Maxwell Law, aur uska crucial Displacement Current term. Result hai Electromagnetic Waves ke liye Wave Equation.

Step 0 — Vocabulary, use karne se pehle draw karo

Kisi bhi calculus se pehle, hum har symbol ko samajh ke kamayenge.

Figure — Speed of light c = 1 - √(ε₀ μ₀)

$E$ aur $B$ ke upar chhoti arrow (jise vector kehte hain) ka matlab bas itna hai ki "is quantity ki ek direction bhi hai, sirf size nahi." Figure dekho: space ke har dot par ek $E$ -arrow (blue) aur ek $B$ -arrow (yellow) hai. Ek field bas itna hi hai — ek rule jo har point par ek arrow pin karta hai.

Definition Do operations jo hum use karenge, words mein

$\frac{\partial}{\partial t}$ — time-derivative: "yeh arrow abhi kitni tezi se badal raha hai?" Ek lamba arrow jo nahi badal raha uska $\partial / \partial t = 0$ hai; ek chhota arrow jo tezi se badh raha hai uska bada hai. Yeh sawaal jawaab deta hai "kya field hil rahi hai?"
$\nabla \times$ — curl: "kya field is point ke aaspaas ghoom rahi hai?" Imagine karo ki arrows mein ek chhota paddlewheel daala; agar woh ghome, toh curl nonzero hai, aur woh spin axis ke saath point karta hai. Yeh jawaab deta hai "kya arrows circulate kar rahe hain?"

Hum sirf yahi do tools choose karte hain kyunki jinse hum shuru kar rahe hain wo do experimental laws literally is form ke statements hain: "ek field ka swirl doosre field ki wobble ke barabar hai." Curl swirl measure karta hai; time-derivative wobble measure karta hai. Ye bilkul sahi instruments hain.

Step 1 — Do starting laws, pictures ke roop mein

Figure dekho: do laws mirror images hain. Har field ka swirl doosri field ki wobble se feed hota hai. Yahi "self-building dominoes" idea hai, ab exactly likha hua. Hum Maxwell's Equations se do aur facts bhi rakhte hain: empty space mein $\nabla \cdot E = 0$ aur $\nabla \cdot B = 0$ (divergence, "kya arrows kisi point se bahar failte hain?", zero hai kyunki koi charge nahi hai jisse failein).

Step 2 — Faraday's law ka curl lo

Figure plan ko ek loop ke roop mein dikhata hai: $E \to$ (uska swirl) $\to B$ ki wobble $\to$ (uska swirl) $\to E$ ki wobble. Us loop ko follow karna hi equation ko sirf $E$ par close kar dega.

Step 3 — Double curl ko suljhao

Ab $\nabla \cdot E = 0$ use karo (Step 1, koi charges nahi): spreading part khatam ho jaata hai.

$\nabla \times (\nabla \times E) = - \nabla^{2} E$

Figure do pieces ko contrast karta hai: left par, arrows ek point se bahar failte hue (divergence — yahan zero, red mein cross out); right par, arrows ek wave crest ki tarah curve karte hue ( $\nabla^{2}$ piece jo hum rakhte hain).

Step 4 — Ampère–Maxwell substitute karo aur $B$ eliminate karo

Do minus signs cancel karo:

$\nabla^{2} E = μ_{0} ε_{0} \frac{\partial ^{2} E}{\partial t ^{2}}$

Figure har surviving symbol ko mark karta hai: blue $\nabla^{2} E$ = "space mein kitna curved," yellow $\partial^{2} / \partial t^{2}$ = "time mein kitna accelerating," aur red $μ_{0} ε_{0}$ = "space-curvature aur time-acceleration ke beech exchange rate."

Step 5 — Wave equation se match karke speed padho

Apni boxed equation ko iske saath rakh do, term ke saath term. Space piece $\nabla^{2}$ , $\partial^{2} / \partial x^{2}$ se match karta hai; time piece match karta hai; toh multiplier bhi match karna chahiye:

$\frac{1}{v ^{2}} = μ_{0} ε_{0} ⟹ v = \frac{1}{μ _{0} ε _{0}} \equiv c$

RECIPROCAL AUR SQUARE ROOT KYUN: constant wave equation ke time side par $1/ v^{2}$ ke roop mein baitha hai. $v$ ko free karne ke liye hum ise paltate hain (reciprocal) aur root lete hain. Figure ek sine wave $f (x - v t)$ ko slide hote dikhaata hai aur slope $v$ label karta hai: bada $μ_{0} ε_{0}$ (stiffer space) $\Rightarrow$ chhota $v$ $\Rightarrow$ dheema crest.

Step 6 — Edge cases (koi gap mat chhodo)

Ek-picture summary

Final figure poori derivation ko ek loop mein dikhata hai: Faraday $B$ ki wobble ko $E$ ke swirl mein convert karta hai; curl + identity ise $\nabla^{2} E$ mein reshape karta hai; Ampère–Maxwell $E$ ki wobble ko $B$ ke swirl mein convert karta hai; substitute karne par loop $E$ par close ho jaata hai, $\nabla^{2} E = μ_{0} ε_{0} \partial^{2} E / \partial t^{2}$ chhod ke; aur Wave Equation se match karke $c = 1/ μ_{0} ε_{0}$ nikalta hai.

Recall Feynman retelling — poora walkthrough seedhe words mein

Nature ke do facts: ek changing magnetic push electric field ko uske aaspaas curl karta hai, aur ek changing electric push magnetic field ko uske aaspaas curl karta hai. Main pehla fact leta hoon aur ise ek baar aur curl karta hoon, jisse main doosra fact swap in kar sakta hoon aur magnetic field ko maths se bilkul bahar nikal sakta hoon. Jo bachta hai woh ek aisi equation hai jo kehti hai ki electric field ki space mein shape field ke time mein acceleration ke barabar hai, do chhote numbers se multiply ki hui jo batate hain ki khali space electricity aur magnetism ke liye kitni stiff hai. Lekin yahi exact form kisi bhi travelling wave ki equation hai, jahan multiplier one-over-speed-squared hota hai. Toh main ise palta hoon aur square root leta hoon — aur speed bahar aa jaati hai: $ε_{0}$ times $μ_{0}$ ke square root ka ulta, jo 300 million metres per second hai. Displacement-current term hatao aur loop kabhi band nahi hota, toh koi light hi nahi hogi. Empty space ki jagah glass ke numbers daalo, aur same machine dheema speed $c / n$ nikaal deti hai. Kisi ne yeh speed choose nahi ki; yeh bas vacuum ki springiness hai.

Visual walkthrough — Speed of light c = 1 - √(ε₀ μ₀)