Speed of light c = 1 - √(ε₀ μ₀): Question bank

1.8.34 · D5 · Physics › Electromagnetism › Speed of light c = 1 - √(ε₀ μ₀)

Yeh deck speed-of-light note ke liye ek rapid-fire trap-spotting deck hai. Har line ek claim, ek error, ek "kyun", ya ek edge case hai. Left side padho, apna jawab ek reason ke saath zor se bolo, phir reveal karo. Sirf "true/false" bolne se kuch nahi milega — reasoning hi sab kuch hai.

Shuru karne se pehle, do words jo aapko parent note se pehle se pata hone chahiye:

permittivity $ε_{0}$ — empty space kitni aasaani se electric field ko allow (support) karta hai.
permeability $μ_{0}$ — empty space kitni aasaani se magnetic field ko allow karta hai.

Neeche diya hua sab test karta hai ki kya aap real fact aur plausible-sounding jhooth mein farq kar sakte ho. Dekho bhi Maxwell's Equations, Displacement Current, Electromagnetic Waves, Refractive Index and n = c/v, aur Wave Equation.

True or false — justify karo

Yahan ek red herring yeh hai ki kai false statements textbook lines jaisi lagti hain. Reveal karne se pehle reason bolo.

Blue light vacuum mein red light se zyada fast travel karti hai.

False —

c = 1/ ε_{0} μ_{0}

mein koi frequency nahi hai, isliye har colour vacuum mein same

c

par travel karta hai. Dispersion (colours ka alag hona) sirf ek medium mein hota hai.

Ek zyada bright (high-intensity) laser beam ek dim beam se zyada fast move karti hai.

False — amplitude/intensity kabhi bhi

c = 1/ ε_{0} μ_{0}

mein appear nahi hota; speed sirf vacuum properties se set hoti hai, fields kitni strong hain isse nahi.

Vacuum mein light ka magnetic field negligibly weak hota hai aur use ignore kiya ja sakta hai.

False —

E = c B

ka matlab hai ki

B

sirf numerically small hai (kyunki

c

bahut bada hai), lekin wave literally

B

ke bina propagate nahi ho sakti; use drop karo aur Faraday's law ke paas relay feed karne ke liye kuch nahi bachta.

Light glass ke andar slow ho jaati hai kyunki glass usse "absorb aur re-emit" karta hai zyada slowly.

Mechanism ke hisaab se broadly true hai, lekin clean statement hai

v = c / ε_{r} μ_{r}

— medium ki badi permittivity (

ε = ε_{r} ε_{0}

) material ki "give" ko change karti hai, wave speed ko kam karti hai.

Formula

c = 1/ ε_{0} μ_{0}

pehle light ki speed measure karke derive kiya gaya tha.

False — beauty iske ulta hai:

ε_{0}

electrostatics se aaya aur

μ_{0}

magnetism se, dono bina kisi optics ke measure kiye gaye, aur unka combination light ki jaani-pehchani speed predict karta hai.

Ek EM wave purely electric field se ban sakti hai jisme koi magnetic field na ho.

False — ek changing

E

B

generate hota hai (Ampère–Maxwell Law) aur vice versa (Faraday's Law of Induction); koi bhi akela travel nahi kar sakta, woh ek doosre ko re-create karte hain.

Ek medium mein jahan

μ_{r} = 1

aur

ε_{r} = 4

hai, refractive index

n = 2

hai.

True —

n = ε_{r} μ_{r} = 4 \cdot 1 = 2

, isliye

v = c /2

; derivation bilkul wahi hai jo vacuum wali hai bas material constants substitute kar dete hain.

ε_{0} μ_{0}

ek speed hai.

False — iske units s/m hain, jo ek slowness hai (seconds per metre). Aapko reciprocal lena hoga:

c = 1/ ε_{0} μ_{0}

ke units m/s hain.

Error dhundho

Har line mein ek planted mistake hai. Use naam do aur correct karo.

"Vacuum mein

J = 0

hai, isliye

\nabla \times B = μ_{0} J = 0

, hence koi bhi EM wave exist nahi kar sakti."

Error hai Displacement Current ko drop karna. Poora law hai

\nabla \times B = μ_{0} J + μ_{0} ε_{0} \partial E / \partial t

;

J = 0

hone par bhi doosra term survive karta hai aur wahi wave launch karta hai.

"Faraday's law ka curl lene par directly

\nabla^{2} E

milta hai."

Pehle aapko double curl

\nabla \times (\nabla \times E)

milta hai; sirf identity

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

aur

\nabla \cdot E = 0

use karne ke baad

- \nabla^{2} E

appear hota hai.

"Wave equation se compare karte hue,

\frac{1}{v ^{2}} = \frac{1}{μ _{0} ε _{0}}

, isliye

v = μ_{0} ε_{0}

Exponent ka sign galat hai. Wave equation deta hai

\frac{1}{v ^{2}} = μ_{0} ε_{0}

, isliye

v = 1/ μ_{0} ε_{0}

"Light ek medium mein slow hoti hai kyunki frequency

f

decrease hoti hai."

Frequency source se fix hoti hai aur medium mein enter karte waqt change nahi hoti; wavelength

λ

(aur isliye

v = f λ

) shrink hoti hai.

"Kyunki

E = c B

aur

c

bada hai, electric field almost saari energy carry karti hai aur magnetic field kuch nahi."

EM wave mein electric aur magnetic energy densities equal hoti hain; factor

c

sirf

E

aur

B

ki numerical values ke beech ek unit-conversion hai, energy split nahi.

ε_{0} μ_{0}

ke units m²/s² hain, jaise ek speed squared."

Iske units s²/m² hain (ulta), isliye

1/ ε_{0} μ_{0}

—

ε_{0} μ_{0}

nahi — m/s mein nikalta hai.

Why questions

Sirf label nahi, mechanism bolo.

Derivation mein

\nabla (\nabla \cdot E)

term kyun vanish ho jaata hai?

Kyunki vacuum mein koi charges nahi hain, isliye Gauss's law deta hai

\nabla \cdot E = 0

; poora gradient term phir zero ho jaata hai, clean Laplacian reh jaata hai.

Light exist karne ke liye displacement current nonzero kyun hona chahiye?

Yeh charge-free vacuum mein

\nabla \times B

ka ek hi source hai; iske bina ek changing

E

koi

B

create nahi kar sakta, self-sustaining

E \leftrightarrow B

relay toot jaata (Electromagnetic Waves).

E

aur

B

bilkul same speed par kyun travel karte hain?

Ampère–Maxwell Law ka curl lene par — Faraday's law ki jagah —

B

ke liye bilkul same wave equation milta hai same

μ_{0} ε_{0}

ke saath, isliye dono fields ek shared

c

par propagate karte hain.

Yeh significant kyun hai ki

ε_{0}

Coulomb's law se aaya aur

μ_{0}

wires ke beech forces se?

Kyunki dono optics ke kisi reference ke bina measure kiye gaye, phir bhi unka combination light ki measured speed reproduce karta hai — woh "coincidence" hi proof hai ki light electromagnetism hai.

Same derivation ek material mein

v = c / n

kyun deta hai?

Aap sirf

ε_{0} \to ε_{r} ε_{0}

aur

μ_{0} \to μ_{r} μ_{0}

swap karo; algebra unchanged rehta hai aur

v = c / ε_{r} μ_{r} = c / n

nikalta hai (dekho Refractive Index and n = c/v).

Light ki speed ko space ki property kyun kaha jaata hai na ki light source ki?

Formula mein sirf

ε_{0}

aur

μ_{0}

hain, jo vacuum ki electric aur magnetic "stiffness" describe karte hain — wave kisne emit ki ya kaise, iske baare mein kuch nahi.

Edge cases

Har quantity ko ek limit tak push karo aur check karo ki formula sahi behave karta hai.

Ek hypothetical vacuum mein

c

kya hoga jisme permittivity double ho,

2 ε_{0}

c^{'} = 1/ 2 ε_{0} μ_{0} = c / 2

; zyada electric "give" relay ko slower banata hai, isliye light lagbhag

30%

slow travel karegi.

Ek medium mein jab

ε_{r} \to \infty

(perfect dielectric),

v

ka kya hoga?

v = c / ε_{r} μ_{r} \to 0

; infinite permittivity wave ko freeze kar degi, ek idealised limit jo koi real material nahi reach karta.

Non-magnetic material (

μ_{r} = 1

) ke liye, kya refractive index

μ_{0}

par depend karta hai?

Speed abhi bhi

ε_{0} μ_{0}

ke through

μ_{0}

par depend karti hai, lekin index

n = ε_{r} μ_{r} = ε_{r}

sirf permittivity tak reduce ho jaata hai, isliye zyaatar optics tables sirf

ε_{r}

list karti hain.

Zero-frequency (static) limit mein, kya koi EM wave hoti hai?

Nahi — ek truly static field mein

\partial E / \partial t = 0

hota hai, isliye displacement current vanish ho jaata hai aur relay kabhi start nahi hota; propagation ke liye changing fields chahiye.

Kya

n < 1

ho sakta hai, matlab

v > c

Phase velocity kuch frequencies ke liye

c

se zyada ho sakti hai (jaise resonances ke paas), lekin koi bhi energy ya signal

c

se fast travel nahi karta, isliye causality safe hai — trap hai phase velocity ko signal speed samajh lena.

Agar

μ_{r}

aur

ε_{r}

dono thode

1

se kam hote, toh kya light vacuum se zyada fast ja sakti hai?

Iske liye

ε_{r} μ_{r} < 1

chahiye; ordinary matter mein

ε_{r} \geq 1

hota hai, isliye real broadband media ke liye yeh nahi hota, though engineered metamaterials narrow bands mein ise mimic kar sakte hain.

Recall Pure deck ka ek-line summary

Yahan har trap chaar jhooton mein se ek hai: (1) $c$ wave par depend karta hai (frequency/intensity) — nahi, sirf vacuum par; (2) aap $B$ ya displacement current drop kar sakte ho — nahi, wahi wave hain; (3) formula $ε_{0} μ_{0}$ hai instead of iska reciprocal — nahi, units check karo; (4) medium mein frequency change hoti hai — nahi, wavelength hoti hai.