Maxwell's equations — integral form, all four

1.8.31 · Physics › Electromagnetism

HUM MAAP KYA RAHE HAIN?

Do geometric quantities har jagah dikhti hain:

1. Gauss's Law for Electricity

HOW banate hain. Ek point charge $q$ banaata hai $E = \frac{q}{4 π ε _{0} r ^{2}} \overset{r}{^}$ . Ek sphere radius $r$ wrap karo: $\oint E \cdot d A = E \cdot (4 π r^{2}) = \frac{q}{4 π ε _{0} r ^{2}} \cdot 4 π r^{2} = \frac{q}{ε _{0}} .$ $r^{2}$ cancel ho jaata hai — yahi inverse-square law ka jaadu hai. Superposition se kisi bhi charge $Q_{e n c}$ ke liye generalize karo:

2. Gauss's Law for Magnetism

Issi liye ek bar magnet mein hamesha N aur S dono poles hote hain: use adha karo toh tumhe do complete magnets milte hain, kabhi ek isolated pole nahi.

3. Faraday's Law of Induction

HOW. EMF hai $\oint E \cdot d ℓ$ (loop ke around work per charge). Experiment: EMF $= - d Φ_{B} / d t$ . Toh:

4. Ampère–Maxwell Law

HOW. Plates ke beech, charge $Q$ accumulate hota hai, $E$ badhta hai, toh $ε_{0} d Φ_{E} / d t$ exactly wire current $I$ ke barabar hota hai. Ise add karo:

Figure — Maxwell's equations — integral form, all four

Worked Examples

Common Mistakes

Flashcards

Gauss's law for electricity (integral form)

\oint_{S} E \cdot d A = Q_{e n c} / ε_{0}

Kisi bhi closed surface se net magnetic flux zero kyun hota hai?

Koi magnetic monopole nahi hote;

B

lines closed loops hain, toh har line jo andar jaati hai bahar bhi nikalti hai.

Faraday's law in integral form

\oint_{C} E \cdot d ℓ = - d Φ_{B} / d t

Faraday's law mein minus sign kya represent karta hai?

Lenz's law — induced EMF flux change ka virodh karta hai.

Ampère–Maxwell law

\oint_{C} B \cdot d ℓ = μ_{0} I_{e n c} + μ_{0} ε_{0} d Φ_{E} / d t

Displacement current kya hai aur uska formula kya hai?

Ek changing electric flux jo

B

produce karta hai current ki tarah;

I_{d} = ε_{0} d Φ_{E} / d t

Kis experiment ne Maxwell ko Ampère's law mein ek term add karne par majbur kiya?

Charging capacitor — plates ke beech koi charge flow nahi karta phir bhi wahan

B

exist karta hai.

Maxwell's equations se EM waves ki speed

c = 1/ μ_{0} ε_{0}

Closed vs open surface: kaun si laws kaun si use karti hain?

Gauss laws closed surfaces use karti hain; Faraday aur Ampère–Maxwell open surfaces use karte hain jo ek loop se bounded hon.

Gauss's law mein, kya surface ke bahar ka charge net flux mein contribute karta hai?

Nahi — uski field lines enter aur exit karti hain, net flux zero deti hain.

Recall Feynman: ek 12-saal ke bachche ko explain karo

Field "lines" ko invisible strings ki tarah imagine karo. Electric strings plus charges se start hoti hain aur minus charges pe khatam hoti hain — toh agar tum ek bag kuch plus charge ke around rakho, zyada strings bahar nikalti hain andar se (Law 1). Magnetic strings ke kabhi ends nahi hote — woh hamesha loops hain — toh ek magnet ke around bag mein equal strings andar jaati hain aur bahar nikalti hain, net zero (Law 2). Agar tum magnetic strings ko wiggle karo, woh electric strings ko ek circle mein dhakelte hain (Law 3, Faraday). Agar tum electric strings ko wiggle karo, woh magnetic strings ko ek circle mein dhakelte hain (Law 4, Ampère–Maxwell). Ek ko wiggle karo, woh doosre ko wiggle karta hai, aur yeh pair ek wave ki tarah nikal jaata hai — woh wave light hai!

Connections

Gauss's Law — $E$ compute karne ke liye symmetry tricks.
Faraday's Law and Lenz's Law — induction detail mein.
Displacement Current — Maxwell ka correction.
Electromagnetic Waves — jahan se $c = 1/ μ_{0} ε_{0}$ aata hai.
Divergence and Curl — inhi laws ka differential form.
Capacitors — charging-capacitor thought experiment.