Noesis
1.8.31Electromagnetism

Maxwell's equations — integral form, all four

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🇮🇳 Yeh note Hinglish mein padho →

WHAT are we even measuring?

Two geometric quantities show up everywhere:

1. Gauss's Law for Electricity

HOW we build it. A point charge qq makes E⃗=q4πε0r2r^\vec E = \dfrac{q}{4\pi\varepsilon_0 r^2}\hat r. Wrap a sphere radius rr: ∮E⃗⋅dA⃗=E⋅(4πr2)=q4πε0r2⋅4πr2=qε0.\oint \vec E\cdot d\vec A = E\cdot(4\pi r^2) = \frac{q}{4\pi\varepsilon_0 r^2}\cdot 4\pi r^2 = \frac{q}{\varepsilon_0}. The r2r^2 cancels — that's the magic of the inverse-square law. Generalize by superposition to any charge QencQ_{enc}:

2. Gauss's Law for Magnetism

This is why a bar magnet always has both N and S poles: snap it in half and you get two complete magnets, never an isolated pole.

3. Faraday's Law of Induction

HOW. EMF is ∮E⃗⋅dℓ⃗\oint \vec E\cdot d\vec\ell (work per charge around the loop). Experiment: EMF =− dΦB/dt= -\,d\Phi_B/dt. So:

4. Ampère–Maxwell Law

HOW. Between the plates, charge QQ accumulates, E⃗\vec E grows, so ε0 dΦE/dt\varepsilon_0\,d\Phi_E/dt exactly equals the wire current II. Add it in:

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

Worked Examples

Common Mistakes

Flashcards

Gauss's law for electricity (integral form)
∮SE⃗⋅dA⃗=Qenc/ε0\oint_S \vec E\cdot d\vec A = Q_{enc}/\varepsilon_0
Why is net magnetic flux through any closed surface zero?
No magnetic monopoles; B⃗\vec B lines are closed loops, so every line in also goes out.
Faraday's law in integral form
∮CE⃗⋅dℓ⃗=− dΦB/dt\oint_C \vec E\cdot d\vec\ell = -\,d\Phi_B/dt
What does the minus sign in Faraday's law represent?
Lenz's law — the induced EMF opposes the change in flux.
Ampère–Maxwell law
∮CB⃗⋅dℓ⃗=μ0Ienc+μ0ε0 dΦE/dt\oint_C \vec B\cdot d\vec\ell = \mu_0 I_{enc} + \mu_0\varepsilon_0\,d\Phi_E/dt
What is displacement current and its formula?
A changing electric flux that produces B⃗\vec B like a current; Id=ε0 dΦE/dtI_d=\varepsilon_0\,d\Phi_E/dt.
Which experiment forced Maxwell to add a term to Ampère's law?
The charging capacitor — no charge flows between plates yet B⃗\vec B exists there.
Speed of EM waves from Maxwell's equations
c=1/μ0ε0c = 1/\sqrt{\mu_0\varepsilon_0}
Closed vs open surface: which laws use which?
Gauss laws use closed surfaces; Faraday & Ampère–Maxwell use open surfaces bounded by a loop.
In Gauss's law, does charge outside the surface contribute to net flux?
No — its field lines enter and exit, giving zero net flux.
Recall Feynman: explain to a 12-year-old

Imagine field "lines" as invisible strings. Electric strings start on plus charges and end on minus charges — so if you put a bag around some plus charge, more strings poke out than in (Law 1). Magnetic strings never have ends — they're always loops — so a bag around a magnet has equal strings going in and out, net zero (Law 2). If you wiggle the magnetic strings, they shove electric strings into a circle (Law 3, Faraday). If you wiggle electric strings, they shove magnetic strings into a circle (Law 4, Ampère–Maxwell). Wiggle one, it wiggles the other, and the pair runs off as a wave — that wave is light!

Connections

Concept Map

used by

used by

used by

used by

source of E flux

zero B flux

induces circulating E

Maxwell's fix

creates

couples with

coupled fields predict

coupled fields predict

Flux through surface

Circulation around loop

Gauss Electricity

Gauss Magnetism

Faraday Induction

Ampere-Maxwell

Enclosed charge

No magnetic monopoles

Changing B flux

Changing E flux

Displacement current

EM waves - light

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, Maxwell ke chaar equations basically do cheezein measure karte hain: flux (kitni field lines kisi surface ke through nikalti hain) aur circulation (field kitna loop mein ghoomti hai). Pehla — Gauss for E — kehta hai ki closed bag ke andar jitna charge hai, utni hi net electric flux bahar nikalti hai. Bahar ka charge count nahi hota kyunki uski lines andar aati hain aur bahar bhi chali jaati hain, net zero.

Doosra — Gauss for B — simply bolta hai magnetic monopole exist nahi karta, isliye magnetic flux har closed surface se zero hoti hai. Magnet ke lines hamesha closed loop banate hain. Teesra — Faraday — changing magnetic flux ek circulating electric field paida karti hai, aur minus sign Lenz's law hai (nature opposition karti hai change ka).

Chautha sabse interesting hai — Ampère–Maxwell. Original Ampère sirf current ko maanta tha, par charging capacitor mein plates ke beech koi charge flow nahi karta phir bhi B⃗\vec B banta hai. Maxwell ne displacement current (ε0 dΦE/dt\varepsilon_0\, d\Phi_E/dt) add ki — yani badalti hui electric flux bhi current jaisa magnetic effect deti hai. Iska sabse bada result: changing EE banata BB, changing BB banata EE, aur ye dono milke wave banke chalte hain speed c=1/μ0ε0c=1/\sqrt{\mu_0\varepsilon_0} pe — yani light! Exam mein closed vs open surface ka dhyaan rakhna aur minus sign mat bhoolna.

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Connections

Gauss's law — integral form, choosing Gaussian surfacesPhysics · 1.8.6Faraday's law — EMF = −dΦ - dtPhysics · 1.8.26Displacement current — Maxwell's addition to Ampere's lawPhysics · 1.8.32Electromagnetic waves — derivation from Maxwell's equationsPhysics · 1.8.33Stokes' theorem — statement, curl-circulation connectionMaths · 4.4.32Gauss's law — integral form, choosing Gaussian surfacesPhysics · 1.8.6Ampere's circuital law — magnetostatic formPhysics · 1.8.23Faraday's law — EMF = −dΦ - dtPhysics · 1.8.26Displacement current — Maxwell's addition to Ampere's lawPhysics · 1.8.32Electromagnetic waves — derivation from Maxwell's equationsPhysics · 1.8.33