Speed of light c = 1 - √(ε₀ μ₀): Worked examples

1.8.34 · D3 · Physics › Electromagnetism › Speed of light c = 1 - √(ε₀ μ₀)

Yeh page speed-of-light note ke liye "har scenario" ka drill hai. Hum koi naya theory introduce nahi karenge — balki hum har tarah ke question ko hit karenge jo yeh ek formula throw kar sakta hai: forward (find $c$ ), backward (find a constant), media, degenerate limits, $E = c B$ amplitude link, aur ek exam twist.

The scenario matrix

Kuch bhi work karne se pehle, yeh cases ka poora map hai. Har worked example ko us cell ke saath tag kiya gaya hai jo wo fill karta hai, taaki tum dekh sako ki coverage complete hai.

#	Case class	What is unknown	Twist / edge
A	Forward — plug in $ε_{0}, μ_{0}$	find $c$	none (baseline)
B	Backward — measured $c$ known	find $μ_{0}$ (or $ε_{0}$ )	inverting the formula
C	Medium — $ε_{r}, μ_{r}$ given	find $v$ and $n$	non-trivial $μ_{r} \neq = 1$
D	Degenerate: $μ_{r} \to 1$	check reduces to $n = ε_{r}$	limiting behaviour
E	Degenerate: constant $\to 0$ or $\to \infty$	what happens to $c$ ?	zero/infinite "give"
F	Amplitude link — $E = c B$	find $B$ from $E$	units-driven
G	Word problem — real world	time / distance	Sun→Earth
H	Exam twist — " $c = ε_{0} μ_{0}$ ?"	spot the error by units	reciprocal trap

Cells covered: A→Ex1, B→Ex2, C→Ex3, D→Ex4, E→Ex5, F→Ex6, G→Ex7, H→Ex8. Har cell ka ek example hai.

Cell A — the forward baseline

Worked example Ex 1. Do constants se

$c$ compute karo Given $ε_{0} = 8.85 \times 1 0^{- 12} F m^{- 1}$ aur $μ_{0} = 4 π \times 1 0^{- 7} T m A^{- 1}$ , find $c$ .

Forecast: aage padhne se pehle guess karo: kya answer $3 \times 1 0^{8}$ ke paas hoga? Kaunsa order of magnitude?

Do constants ko multiply karo. $ε_{0} μ_{0} = (8.85 \times 1 0^{- 12}) (1.2566 \times 1 0^{- 6}) = 1.112 \times 1 0^{- 17} .$ Yeh step kyun? Formula ko root ke neeche product chahiye; pehle ise form karna arithmetic ko ek clean number mein rakhta hai.
Square root lo. $1.112 \times 1 0^{- 17} = 3.335 \times 1 0^{- 9} .$ Yeh step kyun? $ε_{0} μ_{0}$ "slowness" hai (seconds per metre); invert karne se pehle isko chahiye.
Invert karo. $c = \frac{1}{3.335 \times 1 0 ^{- 9}} = 3.00 \times 1 0^{8} m s^{- 1} .$ Yeh step kyun? $c = 1/ ε_{0} μ_{0}$ — reciprocal slowness ko speed mein badalta hai.

Verify: units. $ε_{0} μ_{0}$ units $s^{2} m^{- 2}$ carry karta hai, toh iska root $s m^{- 1}$ hai (ek slowness), aur $1/ slowness = m s^{- 1}$ — ek speed. ✓ Numerically yeh optically measured $c$ se match karta hai. Woh match hi proof hai ki light electromagnetic hai.

Cell B — formula ko backward run karna

Worked example Ex 2.

$c$ aur $ε_{0}$ se $μ_{0}$ recover karo Maan lo kisi experimenter ne $c = 3.00 \times 1 0^{8} m s^{- 1}$ measure kiya aur $ε_{0} = 8.85 \times 1 0^{- 12} F m^{- 1}$ measure kiya lekin $μ_{0}$ ki koi value nahi hai. $μ_{0}$ find karo.

Forecast: kya answer $1.26 \times 1 0^{- 6}$ ke paas aana chahiye? Exponent guess karo.

$c = \frac{1}{ε _{0} μ _{0}}$ se start karo aur dono sides ko square karo. $c^{2} = \frac{1}{ε _{0} μ _{0}} .$ Yeh step kyun? Squaring awkward square root ko hata deta hai taaki hum $μ_{0}$ ke liye linearly solve kar sakein.
$μ_{0}$ ke liye rearrange karo. $μ_{0} = \frac{1}{ε _{0} c ^{2}} .$ Yeh step kyun? $μ_{0}$ akela unknown hai; use algebraically isolate karo.
Plug in karo. $μ_{0} = \frac{1}{( 8.85 \times 1 0 ^{- 12} ) ( 3.00 \times 1 0 ^{8} ) ^{2}} = \frac{1}{( 8.85 \times 1 0 ^{- 12} ) ( 9.0 \times 1 0 ^{16} )} = 1.255 \times 1 0^{- 6} .$ Yeh step kyun? Do measured numbers ki direct substitution.

Verify: accepted $μ_{0} = 4 π \times 1 0^{- 7} = 1.2566 \times 1 0^{- 6} T m A^{- 1}$ . Hamara recovered value teen figures tak agree karta hai. ✓ Yeh historical logic ulta hai: $(c, ε_{0}, μ_{0})$ mein se koi bhi do teesre ko fix kar dete hain.

Cell C — ek real material ke andar light

Worked example Ex 3. Ek magnetic dielectric mein speed aur index

Ek hypothetical medium mein relative permittivity $ε_{r} = 4.0$ aur relative permeability $μ_{r} = 2.25$ hai. Uska refractive index $n$ aur wave speed $v$ find karo.

Forecast: dono constants 1 se upar hain, toh kya light vacuum se fast hogi ya slow? Roughly kitna?

Derivation mein vacuum constants ki jagah material wale use karo: $v = \frac{1}{ε μ}$ with $ε = ε_{r} ε_{0}, μ = μ_{r} μ_{0}$ . Yeh step kyun? Wave equation ek medium mein same hoti hai; sirf constants badle hain, isliye $v$ ki form unchanged hai. Dekho Refractive Index and n = c/v.
Vacuum part ko factor out karo. $v = \frac{1}{ε _{r} μ _{r}} \cdot \frac{1}{ε _{0} μ _{0}} = \frac{c}{ε _{r} μ _{r}} .$ Yeh step kyun? Isse $n = ε_{r} μ_{r}$ cleanly expose hota hai aur jaana-maana $c$ reuse hota hai.
$n$ compute karo. $n = 4.0 \times 2.25 = 9.0 = 3.0.$ Yeh step kyun? $n$ woh akela number hai jo slow-down factor batata hai.
$v$ compute karo. $v = \frac{3.00 \times 1 0 ^{8}}{3.0} = 1.00 \times 1 0^{8} m s^{- 1} .$ Yeh step kyun? $v = c / n$ index ko directly speed mein badal deta hai.

Verify: dono $ε_{r}, μ_{r} > 1$ ⇒ medium vacuum se zyada "sluggishly give" karta hai ⇒ $v < c$ . Indeed $1 \times 1 0^{8} < 3 \times 1 0^{8}$ . ✓ Aur $n > 1$ jaise ek real passive medium ke liye hona chahiye.

Cell D — degenerate limit $μ_{r} \to 1$

Worked example Ex 4. Non-magnetic glass (

$μ_{r} = 1$ ) Glass mein $ε_{r} \approx 2.25$ hai aur, non-magnetic hone ki wajah se, $μ_{r} \approx 1$ . Dikhao ki index formula $n = ε_{r}$ mein collapse ho jata hai aur $v$ find karo.

Forecast: glass ke liye $n$ guess karo — kya yeh 1.5 ke paas hai?

$n = ε_{r} μ_{r}$ se start karo aur $μ_{r} = 1$ set karo. $n = ε_{r} \cdot 1 = ε_{r} .$ Yeh step kyun? Zyaatar optical materials non-magnetic hain, isliye yeh woh case hai jo tum 99% time dekhte ho; yeh worth hai ki general formula ko isme reduce hote dekha jaaye, na ki ek alag rule memorise kiya jaaye.
Evaluate karo. $n = 2.25 = 1.5.$ Yeh step kyun? Ek remaining constant plug in karo.
Speed. $v = \frac{c}{n} = \frac{3.00 \times 1 0 ^{8}}{1.5} = 2.00 \times 1 0^{8} m s^{- 1} .$ Yeh step kyun? Phir se $v = c / n$ .

Verify: yeh textbook glass value hai ( $n = 1.5$ , $v = 2 \times 1 0^{8}$ ). ✓ Note karo ki limit smooth hai: jab $μ_{r} \to 1$ toh full formula aur reduced formula same number dete hain, isliye boundary par kuch "break" nahi hota.

Cell E — zero aur infinite "give"

Worked example Ex 5. Agar koi constant

$0$ ya $\infty$ ho jaaye toh? Purely ek limiting thought-experiment ke taur par, $c = 1/ ε_{0} μ_{0}$ ko examine karo jab ek constant $0$ ya $\infty$ ki taraf dial kiya jaata hai. $c$ ka kya hota hai? (Koi plugging-in nahi — limits ke baare mein reasoning.)

Forecast: agar space fields ko zero resistance offer kare ( $ε_{0} μ_{0} \to 0$ ), toh kya tum expect karte ho ki light infinitely fast ho ya infinitely slow?

Case $ε_{0} μ_{0} \to 0^{+}$ (space ki "stiffness" vanish ho rahi hai). $c = \frac{1}{ε _{0} μ _{0}} ⟶ \frac{1}{0 ^{+}} = + \infty.$ Yeh step kyun? Denominator zero ki taraf upar se shrink hota hai, isliye uska reciprocal blow up karta hai. Physically: agar space fields ko bilkul resist na kare, toh field "dominoes" infinitely fast girenge.
Case $ε_{0} μ_{0} \to \infty$ (infinitely "stiff" space). $c = \frac{1}{\infty} ⟶ 0^{+} .$ Yeh step kyun? Root ke neeche ek bahut bada product reciprocal ko zero ki taraf collapse kar deta hai: infinitely stiff space wave ko freeze kar deta.
Signs ka reality check: $ε_{0} > 0$ aur $μ_{0} > 0$ hamesha (yeh ek physical "give" measure karte hain, negative nahi), isliye $ε_{0} μ_{0} > 0$ strictly aur root real hai. Koi aisa case nahi hai jahan $c$ imaginary ya negative ho. Yeh step kyun? Yeh sign question close karta hai — "all quadrants check karne" ka analogue. Har physical input $c$ ko real aur positive rakhta hai.

Verify: monotonic aur consistent — zyada stiffness ⇒ slower light; kam stiffness ⇒ faster. Jo finite real value hum actually measure karte hain ( $3 \times 1 0^{8}$ ) woh in extremes ke beech hai kyunki vacuum mein finite, positive stiffness hai. ✓

Cell F — $E = c B$ amplitude link

Worked example Ex 6. Sunlight ka magnetic field

Ek plane light wave ka electric-field amplitude $E_{0} = 900 V m^{- 1}$ hai. Magnetic-field amplitude $B_{0}$ find karo.

Forecast: kyunki $c$ bahut bada hai, kya tum expect karte ho ki $B_{0}$ tesla mein ek bada ya tiny number hoga?

Electromagnetic Waves se plane-wave relation use karo: $E = c B$ , toh $B_{0} = E_{0} / c$ . Yeh step kyun? Ek single EM wave mein $E$ aur $B$ Maxwell's equations se lock hain; ratio exactly $c$ hai, koi free choice nahi.
Substitute karo. $B_{0} = \frac{900}{3.00 \times 1 0 ^{8}} = 3.0 \times 1 0^{- 6} T .$ Yeh step kyun? Direct division; factor $c \approx 3 \times 1 0^{8}$ number ko enormously shrink karta hai.

Verify: units. $E_{0} / c$ mein $\frac{V m ^{- 1}}{m s ^{- 1}} = V s m^{- 2} = T$ (kyunki $1 T = 1 V s m^{- 2}$ ). ✓ Aur $3 μ T$ Earth ke field ke comparable hai — light ka magnetic part SI numbers mein genuinely tiny hai, exactly kyunki yeh $c$ se divide hota hai.

Cell G — real-world word problem

Worked example Ex 7. Sunlight humtak pahunchne mein kitna time lagta hai?

Sun $1.50 \times 1 0^{11} m$ door Earth se hai. Sunlight vacuum se travel karti hai. Light pahunchne mein kitne seconds — aur minutes — lagte hain?

Forecast: compute karne se pehle minutes mein guess karo. 10 se zyada ya kam?

Vacuum mein light $c$ par move karti hai; time $=$ distance $/$ speed. $t = \frac{d}{c} = \frac{1.50 \times 1 0 ^{11}}{3.00 \times 1 0 ^{8}} .$ Yeh step kyun? Uniform speed ⇒ simplest kinematics; ek baar $c$ jaante hain toh $μ_{0}, ε_{0}$ ki zaroorat nahi.
Seconds compute karo. $t = 5.00 \times 1 0^{2} s = 500 s .$ Yeh step kyun? Powers of ten divide karo: $1 0^{11} /1 0^{8} = 1 0^{3}$ , aur $1.5/3.0 = 0.5$ .
Minutes mein convert karo. $t = \frac{500}{60} = 8.33 min .$ Yeh step kyun? Minutes is famous "eight-minute" fact ke liye intuitive unit hai.

Verify: well-known answer hai "about 8 minutes 20 seconds." $8.33 min = 8 min 20 s$ . ✓ Units: $m / (m s^{- 1}) = s$ . ✓

Cell H — exam twist (trap pakdo)

Worked example Ex 8. Kya

$c = ε_{0} μ_{0}$ kabhi sahi hai? Ek test formula $c = ε_{0} μ_{0}$ offer karta hai. $c$ compute kiye bina, sirf units se decide karo ki yeh possibly ek speed ho sakta hai ya nahi, phir fix batao.

Forecast: kya raw (non-reciprocal) form metres-per-second deta hai, ya uska opposite?

Ex 1 se yaad karo ki $ε_{0} μ_{0}$ ke units $s^{2} m^{- 2}$ hain. Yeh step kyun? Units bina kisi arithmetic ke poora question decide karte hain — yeh sabse cheap possible check hai.
Root ko as written lo. $ε_{0} μ_{0} has units s^{2} m^{- 2} = s m^{- 1} .$ Yeh step kyun? $s m^{- 1}$ ek slowness hai (seconds per metre), speed ka reciprocal.
Numerically yeh $3.335 \times 1 0^{- 9} s m^{- 1}$ hai — "speed" ke taur par absurd. Yeh step kyun? Dikhata hai ki magnitude bhi galat hai, sirf dimension nahi.
Fix: invert karo. $c = 1/ ε_{0} μ_{0} = 3.00 \times 1 0^{8} m s^{- 1}$ . Yeh step kyun? Sirf reciprocal ke units $m s^{- 1}$ hain.

Verify: offered form $3.335 \times 1 0^{- 9}$ deta hai, correct form $3.00 \times 1 0^{8}$ deta hai; yeh reciprocals hain, $ε_{0} μ_{0}$ ke factor se differ karte hain. ✓ Units ne error ko pakad liya pehle koi number trust karne se.

Recall Matrix par self-test

Kaun sa cell formula invert karne ko kehta hai? ::: Cell B (Ex 2) — $μ_{0}$ ke liye solve karo. Kaun sa cell dikhata hai ki general index $n = ε_{r}$ mein reduce hota hai? ::: Cell D (Ex 4), jab $μ_{r} \to 1$ . $c$ kabhi imaginary kyun nahi ho sakta? ::: Kyunki $ε_{0} μ_{0} > 0$ hamesha (dono positive "gives" hain), isliye root real hai (Ex 5). Kaunsa ek check $c = ε_{0} μ_{0}$ error instantly expose karta hai? ::: Units — raw form s/m hai, ek slowness (Ex 8).