Electromagnetic waves — derivation from Maxwell's equations: Question bank

1.8.33 · D5 · Physics › Electromagnetism › Electromagnetic waves — derivation from Maxwell's equations

Definition Do "field-shape" questions jo tum har trap mein use karoge

Yahaan $F$ kisi bhi field arrow ki jagah hai — ise electric field $E$ (jo positive charge ko push karta hai) ya magnetic field $B$ (jo moving charge ko curve karta hai) samjho. Neeche sab kuch ek baar $F$ ke liye bataya gaya hai, phir naam se $E$ aur $B$ par apply hoga.

$\nabla \cdot F$ (the divergence) puchta hai "is point par, kya arrows kisi source se bahar phail rahe hain ya kisi sink mein ghus rahe hain?" Zero ka matlab hai arrows seedha guzar jaate hain — yahan koi shuruat nahi, koi ant nahi. Toh $\nabla \cdot E = 0$ ka matlab hai wahan koi charges nahi hain, aur $\nabla \cdot B = 0$ ka matlab hai magnetic field lines kabhi bhi shuru ya khatam nahi hoti.
$\nabla \times F$ (the curl) puchta hai "is point par, kya arrows kisi loop ke around swirl kar rahe hain?" Ek badalta hua partner field hi is swirl ko force karta hai: badalta hua $B$ , $E$ ko swirl karta hai, aur badalta hua $E$ , $B$ ko swirl karta hai.

Definition Notation jo tumhe line one se pehle pata honi chahiye

Is poore page par teen shorthand baar baar aate hain. Pehle inhe padho.

Hat $^$ ka matlab "unit vector" hai. Kisi letter par chota sa hat, jaise $\hat{E}$ , $\hat{B}$ , $\hat{k}$ , matlab hai "ek arrow jiska length bilkul $1$ hai aur jo us cheez ki direction mein point karta hai." Toh $\hat{E}$ sirf woh direction hai jis taraf electric field point karta hai, uski size hata ke; $\hat{k}$ woh pure direction hai jis taraf wave travel karti hai. Ek puri lambai ka field arrow imagine karo, phir uski heading rakhte hue use length one tak shrink karo.
Laplacian $\nabla^{2}$ ka matlab hai "ek field kitna apne neighbourhood average se bulge karta hai." Kisi field ke har component ke liye yeh second space-derivatives ko add karta hai, $\nabla^{2} f = \partial_{x}^{2} f + \partial_{y}^{2} f + \partial_{z}^{2} f$ . Intuitively: kya yahan ki value immediately aaspaas ke points ke average se zyada hai ya kam? Ek wave equation exactly ek rule hai jo is spatial bulge ko time mein field ki acceleration se link karta hai.
Imaginary unit $i$ ka matlab woh number hai jiske liye $i^{2} = - 1$ hai. Yeh kisi bhi measurement se correspond nahi karta — neeche diya box dekho. Yeh $e^{i (k x - ω t)}$ mein sirf bookkeeping ke liye saath chalti hai.

Definition Is page par har symbol, simple shabdon mein

Koi bhi symbol yahan se dekhlo kisi trap ko padhne se pehle.

$E$ — the electric field (units V/m), ek point par electric push ka arrow; $\hat{E}$ uski unit-vector direction hai.
$B$ — the magnetic field (units tesla, T), ek point par magnetic influence ka arrow; $\hat{B}$ uski direction hai.
$ρ$ (rho) — the charge density: ek tiny volume mein kitna electric charge baitha hai. Vacuum mein $ρ = 0$ .
$J$ — the current density: ek tiny area se har second mein kitna charge flow karta hai. Vacuum mein $J = 0$ .
$ε_{0} \approx 8.854 \times 1 0^{- 12} F/m$ — the permittivity of free space, electric (electrostatic) force experiments se measure kiya; yeh set karta hai ki ek charge kitna strong electric field banata hai.
$μ_{0} = 4 π \times 1 0^{- 7} T\cdotm/A$ — the permeability of free space, magnetic force experiments se measure kiya; yeh set karta hai ki ek current kitna strong magnetic field banata hai.
$c = 1/ μ_{0} ε_{0} \approx 3.00 \times 1 0^{8}$ m/s — vacuum mein speed of light.
$k$ — the wave number, $k = 2 π / λ$ : space ke har metre mein kitne radians ka wiggle fit hota hai. $k$ woh arrow hai jo wave ke travel direction mein point karta hai; $\hat{k}$ woh direction unit vector ke roop mein hai.
$ω$ (omega) — the angular frequency, $ω = 2 π f$ : har second mein kitne radians ka wiggle hota hai.
$E_{0}, B_{0}$ — electric aur magnetic wiggles ki amplitudes (peak heights).
$λ$ (lambda) — the wavelength; $f$ — Hz mein ordinary frequency; $c = f λ$ se related.
$\nabla^{2}$ — the Laplacian, upar wale box mein define kiya.

Divergence aur curl ke questions dhyan mein rakho — neeche lagbhag har trap asal mein is baare mein hai ki unme se kaun sa switched on hai.

True or false — justify

TF1. Vacuum mein bina charges ke,

E

field lines phir bhi kisi cheez par shuru aur khatam ho sakti hain.

False.

\nabla \cdot E = 0

kehta hai koi sources ya sinks nahi hain, toh lines kabhi start ya stop nahi hoti — ya toh woh khud par close hoti hain ya infinity tak jaati hain.

TF2. Displacement current term hatane ke baad bhi ek wave equation bachti hai, bas slower wave ke liye.

False.

μ_{0} ε_{0} \partial_{t} E

ke bina, Ampère's law vacuum mein

\nabla \times B = 0

deta hai; Step 3 phir

0

deta hai aur second time-derivative bilkul gayab ho jaata hai — kisi bhi speed ki koi wave nahi.

TF3.

c = 1/ μ_{0} ε_{0}

ka relation ek light pulse ko time karke measure kiya gaya tha.

False. Yeh wave equation ke coefficient se nikalta hai;

μ_{0}

(magnetic force experiments se) aur

ε_{0}

(electrostatic experiments se) bina kisi light ke measure kiye jaate hain, phir bhi unka combination light ki speed predict karta hai.

TF4. Kyunki

E_{0} = c B_{0}

aur

c

bahut bada hai, electric field bahut zyada energy carry karta hai magnetic field se.

False. Energy densities equal hain,

\frac{1}{2} ε_{0} E_{0}^{2} = \frac{1}{2 μ _{0}} B_{0}^{2}

; bada numerical ratio purely ek SI-unit artefact hai, koi physical imbalance nahi.

TF5. Vacuum mein ek EM wave ka

E

uski travel direction ke along point kar sakta hai.

False.

\nabla \cdot E = 0

force karta hai

ik E_{0 x} = 0

, toh

k

(travel direction) ke along component zero ho jaata hai — wave strictly transverse hai.

TF6. Plane wave mein

E

aur

B

9 0^{\circ}

out of phase hain, jaise ek reactive circuit mein current aur voltage.

False.

e^{i (k x - ω t)}

(real part

cos (k x - ω t)

) ke saath Faraday's law deta hai

k E_{0} = ω B_{0}

bina kisi relative phase ke, toh

E

aur

B

saath peak karte hain aur saath zero hote hain — in phase hain.

TF7. Curl-of-curl identity hamesha

\nabla^{2} E = μ_{0} ε_{0} \partial_{t}^{2} E

deti hai bina kisi extra term ke.

False. Generally

\nabla (\nabla \cdot E)

term bachta hai; sirf charge-free condition

\nabla \cdot E = 0

use khatam karta hai. Charged medium ke andar yeh rehta hai aur cheezein complicated ho jaati hain.

TF8. Dono

E

aur

B

usi wave equation ko usi speed

c

ke saath satisfy karte hain.

True. Ampère–Maxwell ka curl lete hue aur

\nabla \cdot B = 0

use karke

E

ki derivation exactly mirror hoti hai, deta hai

\nabla^{2} B = μ_{0} ε_{0} \partial_{t}^{2} B

TF9. Polarization refer karta hai ki

E

B

mein se kaun bada hai.

False. Polarization woh direction hai jismein

E

k

ke transverse plane mein oscillate karta hai; dekho Polarization. Amplitude size ek alag matter hai aur

E_{0} = c B_{0}

hamesha hota hai.

TF10. Linear polarization (E ek fixed line ke along oscillate karta hai) woh akela type hai jo ek EM wave mein ho sakta hai.

False. Agar do perpendicular transverse components

9 0^{\circ}

out of phase aur equal hain, toh

E

ka tip ek circle trace karta hai — circular polarization; unequal amplitudes ya doosre phases ek ellipse trace karte hain — elliptical polarization. Linear sirf zero phase offset ka special case hai.

Spot the error

SE1. "

\nabla \times E = - \partial_{t} B

se, hum

B

ko Gauss's law

\nabla \cdot B = 0

se substitute karte hain."

Tum ek divergence equation se

B

extract nahi kar sakte — woh sirf kehta hai

B

ke koi sources nahi hain. Woh substitution jo loop close karti hai woh Ampère–Maxwell (4) use karta hai, jo

\nabla \times B

supply karta hai.

SE2. "Kyunki

E_{0} = c B_{0}

hai, SI units mein

E

wave aur

B

wave alag physical waves hain."

Woh ek wave hain.

E

aur

B

phase mein locked hain, orthogonal hain, ek single ripple ki tarah move karte hain; factor

c

sirf unki units ke beech numerical bridge hai.

SE3. "

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

identity se."

Identity hai

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

; pehla term sirf tab drop kar sakte ho jab

\nabla \cdot E = 0

invoke karo. Use bina yeh bataye

- \nabla^{2} E

likhna ek key physics step skip karta hai.

SE4. "Ek plane wave

e^{i (k x - ω t)}

kisi bhi

ω

aur

k

ke liye wave equation solve karta hai."

Sirf tabhi jab dispersion relation

ω = c k

hold kare. Plug in karne par milta hai

- k^{2} = - μ_{0} ε_{0} ω^{2}

, jo

ω / k = c

pin karta hai; doosre pairs solutions nahi hain.

SE5. "

\hat{E} \times \hat{B}

backward point karta hai, wave ke travel ke opposite direction mein."

Right-handed triad ke liye

\hat{E} \times \hat{B} = \hat{k}

hota hai, jo travel ki direction ke along point karta hai — wahi direction jismein Poynting vector

S = \frac{1}{μ _{0}} E \times B

energy carry karta hai.

SE6. "Vacuum mein

J = 0

hai, toh Ampère–Maxwell reduce ho jaata hai

\nabla \times B = 0

mein."

Displacement Current term

μ_{0} ε_{0} \partial_{t} E

tab bhi rehta hai jab current density

J = 0

ho. Use drop karna exactly woh mistake hai jo light ko equations se mita deti hai.

SE7. "Step 3 mein

\partial_{t}

aur

\nabla \times

ko swap karne ke liye ek extra physical assumption chahiye."

Ise sirf fields ka smooth hona chahiye (mixed partials commute karte hain), jo ek mathematical regularity condition hai, naya physics nahi.

Why questions

WHY1. Hum Faraday's law ka curl doosri baar kyun lete hain instead of directly coupled pair solve karne ke?

Doosra curl

\nabla \times B

laata hai, jise Ampère–Maxwell humein

\partial_{t} E

se replace karne deta hai,

B

eliminate karta hai aur do coupled first-order equations ko sirf

E

mein ek closed second-order equation mein collapse karta hai.

WHY2. Charge-free condition

ρ = 0

ek clean wave equation paane ke liye kyun matter karti hai?

Yeh

\nabla \cdot E = 0

set karta hai, jo curl-of-curl identity se

\nabla (\nabla \cdot E)

term ko annihilate karta hai, pure Laplacian

\nabla^{2} E

chodta hai jo standard Wave Equation template ko chahiye.

WHY3. Yeh "stunning" kyun hai ki

c

μ_{0}

aur

ε_{0}

se aata hai?

Koi bhi constant light use karke measure nahi kiya gaya tha —

μ_{0}

magnetic forces se,

ε_{0}

electric forces se — phir bhi unka combination light ki speed ke equal nikla, ye reveal karta hai ki light ek electromagnetic phenomenon hai.

WHY4. EM waves transverse kyun honi chahiye jabki sound longitudinal ho sakti hai?

Sound ek pressure oscillation hai bina kisi divergence-free constraint ke; EM waves

\nabla \cdot E = 0

follow karte hain, jo

k

ke along kisi bhi oscillation component ko forbid karta hai, toh wiggle sideways honi chahiye.

WHY5. Light ke sab colours — radio se gamma tak — vacuum mein ek hi speed

c

kyun share karte hain?

c = 1/ μ_{0} ε_{0}

mein koi frequency nahi hai, toh Electromagnetic Spectrum mein har frequency ek identical vacuum speed par travel karti hai; sirf

λ

badalta hai

c = f λ

se.

WHY6. Faraday's law

B

E

ke perpendicular hone ke liye kyun force karta hai?

E

\overset{y}{^}

along aur travel

\overset{x}{^}

along hone par, curl

\nabla \times E

\overset{z}{^}

along point karta hai; kyunki yeh

- \partial_{t} B

ke equal hai, field

B

bhi

\overset{z}{^}

ke along rehna chahiye —

E

ke perpendicular.

WHY7. Displacement current ko woh ingredient kyun kaha jaata hai "jo light ko possible banata hai"?

Yeh akela term hai jo vacuum mein badlte

E

ko curling

B

se wapas couple karta hai; iske bina feedback loop (E wiggles B, B wiggles E) toot jaata hai aur self-sustaining ripple exist nahi kar sakta.

WHY8. Poynting vector

S = \frac{1}{μ _{0}} E \times B

travel direction ke along kyun point karta hai?

Kyunki right-handed triad ke liye

\hat{E} \times \hat{B} = \hat{k}

hai, toh cross product direction

\hat{k}

inherit karta hai — energy usi taraf stream karti hai jis taraf wave move karti hai.

Edge cases

EC1. Wave equation ek static (time-independent) field ke liye kya kehta hai jab

\partial_{t} = 0

ho?

Yeh reduce ho jaata hai

\nabla^{2} E = 0

mein, Laplace's equation — ek valid electrostatic field, lekin koi propagating wave nahi, kyunki ek wave ke liye time-derivative term ka nonzero hona zaruri hai.

EC2. Agar

B_{0} = 0

har jagah ho, toh kya ek pure

E

wave vacuum mein travel kar sakti hai?

Nahi. Ek badalta hua

E

necessarily Ampère–Maxwell se ek

B

curl up karta hai, aur wahi

B

hai jo

E

ko regenerate karta hai;

B

ko khatam karo toh feedback khatam, toh koi wave survive nahi karti.

EC3.

E_{0} / B_{0} = c

kya hoga agar wave ek aisi medium mein enter kare jahan speed

v < c

ho?

Ratio ban jaata hai

E_{0} / B_{0} = v

c

nahi, kyunki amplitude relation

ω / k

se aati hai, jo us medium mein actual phase speed ke equal hoti hai.

EC4.

ω \to 0

ke limit mein (ek wave jo barely wiggle kare), wavelength ka kya hoga?

λ = 2 π c / ω

se, wavelength infinity tak diverge ho jaati hai; truly non-oscillating field static limit hai — "wave" itni spread out ho jaati hai ki woh wave nahi rehti.

EC5. Kya derivation badal jaati hai agar wave

+ x

ki jagah

- x

ke along travel kare?

Koi physics nahi badlta;

e^{i (- k x - ω t)}

use karo ya

k < 0

lo. Triad flip hoti hai taki

\hat{E} \times \hat{B}

phir bhi nayi travel direction ke along point kare, aur

E_{0} = c B_{0}

unchanged rehta hai.

EC6. Kya

E_{0} = c B_{0}

ek vector equation hai?

Nahi — yeh scalar amplitudes ko relate karta hai. Vectors

E

aur

B

alag (perpendicular) directions mein point karte hain, toh tum

E_{0} = c B_{0}

nahi likh sakte.

EC7. Kya hoga agar hum

+ x

mein jaane wali wave ke liye

E

\overset{y}{^}

ki jagah

\overset{z}{^}

ke along choose karein?

Bilkul allowed hai — yeh sirf ek alag linear polarization hai. Axes relabel karke derivation identical hai, aur

B

us remaining transverse axis ke along aa jaata hai jo right-handed triad complete karta hai.

EC8. Circular aur elliptical polarization mein unke beech ki boundary par kya distinguish karta hai?

Dono do perpendicular transverse components se aate hain jinka phase offset hota hai; jab amplitudes equal hoon aur offset exactly

9 0^{\circ}

ho toh

E

ka tip ek perfect circle trace karta hai (circular), aur koi bhi doosra amplitude/phase combination ek ellipse trace karta hai (elliptical) — circular symmetric special case hai.

Visual aids

Figure — Electromagnetic waves — derivation from Maxwell's equations

Recall Har trap ki ek-line summary

Divergence-zero ⇒ transverse aur sourceless; displacement current ⇒ wave exist karti hai; curl-of-curl ⇒ wave equation; in-phase orthogonal triad $E_{0} = c B_{0}$ ke saath, equal energy, aur energy flow $S = \frac{1}{μ _{0}} E \times B$ , $\hat{k}$ ke along; polarization (linear/circular/elliptical) woh path hai jo $E$ ka tip trace karta hai; $c$ frequency-independent do static constants se.