Magnetic flux Φ = ∫B·dA

What is flux, exactly?

Why a dot product? (Derivation from scratch)

We want to count field lines through area. Start with the simplest case and build up.

Step 1 — Field perpendicular to a flat area. If $\mathbf{B}$ is uniform and points straight through a flat area $A$ (field $\perp$ surface, i.e. parallel to the normal), every line passes through. The number of lines $\propto B \times A$ . So $\Phi = BA.$ Why this step? Field-line density represents $B$ ; lines through a patch = density × area.

Step 2 — Tilt the surface. Now tilt the area by angle $\theta$ between $\mathbf{B}$ and the surface normal $\hat{\mathbf{n}}$ . The surface no longer "faces" the field head-on. The effective area that the field sees is the projection $A\cos\theta$ . $\Phi = BA\cos\theta.$ Why this step? Only the component of area facing the field catches lines. A surface tilted edge-on ( $\theta = 90^\circ$ ) catches zero lines because lines slide along it.

Step 3 — Recognize the dot product. $BA\cos\theta$ is exactly $\mathbf{B}\cdot\mathbf{A}$ (with $\mathbf{A}=A\hat{\mathbf n}$ ). So for uniform field, flat surface: $\boxed{\Phi = \mathbf{B}\cdot\mathbf{A} = BA\cos\theta}$

Step 4 — Field varies / surface curves. Chop the surface into tiny patches $d\mathbf{A}$ so small that $\mathbf{B}$ is essentially constant over each. Add up each patch's contribution $\mathbf{B}\cdot d\mathbf{A}$ and take the limit → integral: $\Phi = \int_S \mathbf{B}\cdot d\mathbf{A}.$ Why this step? Calculus = "slice into easy pieces, sum, take limit." The general formula is just the uniform case applied locally.

Formula Summary of cases

Situation	Formula
Uniform $\mathbf{B}$ , flat surface, normal at angle $\theta$	$\Phi = BA\cos\theta$
Uniform $\mathbf{B}$ , surface faces field ( $\theta=0$ )	$\Phi = BA$
Uniform $\mathbf{B}$ , surface edge-on ( $\theta=90^\circ$ )	$\Phi = 0$
General field/surface	$\Phi = \int_S \mathbf{B}\cdot d\mathbf{A}$

Gauss's law for magnetism (a free bonus)

Worked examples

Worked example 4 — Non-uniform field (integral needed)

Field $\mathbf{B} = (0.30\, x)\,\hat{\mathbf z}\ \text{T}$ ( $x$ in metres) through a square in the $xy$ -plane, $0\le x\le 2$ , $0\le y\le 1$ . Normal is $\hat{\mathbf z}$ .

$\mathbf{B}\cdot d\mathbf{A} = 0.30\,x\,dA$ since field $\parallel$ normal. Why? Both along $\hat z$ , dot product = product of magnitudes.
$\Phi = \int_0^1\!\!\int_0^2 0.30\,x\, dx\, dy = 0.30 \cdot \left[\tfrac{x^2}{2}\right]_0^2 \cdot [y]_0^1 = 0.30 \cdot 2 \cdot 1 = 0.60\text{ Wb}$ .
Why integrate? $B$ changes with $x$ , so no single $BA$ works — we sum strip by strip.

Common mistake Steel-manning the common errors

Mistake A: Using the angle between $\mathbf{B}$ and the surface plane. Why it feels right: "Flux flows through the surface, so use the angle with the surface." Sounds natural. The fix: Flux depends on the normal. Always use $\theta$ between $\mathbf{B}$ and $\hat{\mathbf n}$ . If unsure, sanity-check: face-on ⇒ max flux ( $\theta=0$ ); edge-on ⇒ zero ( $\theta=90^\circ$ ).

Mistake B: "More field always means more flux." Why it feels right: Stronger field → more lines. The fix: Orientation matters equally. A huge field edge-on gives zero flux.

Mistake C: Thinking flux through a closed surface can be nonzero. Why it feels right: By analogy with electric charge enclosed (Gauss's law for $E$ ). The fix: No magnetic monopoles → every line that enters leaves → $\oint \mathbf{B}\cdot d\mathbf{A}=0$ .

Mistake D: Forgetting $N$ turns. For a coil of $N$ turns, the flux linkage is $N\Phi$ — each turn catches the flux. EMF uses $N\Phi$ .

Recall Feynman: explain it to a 12-year-old

Imagine rain falling straight down and you hold a paper ring to catch raindrops.

Hold the ring flat (facing up) → tons of drops fall through. That's big flux.
Tilt it → fewer drops pass through.
Hold it sideways (edge into the rain) → no drops go through. That's zero flux. The rain is the magnetic field, the ring is your surface, and flux = how many drops sneak through. Two things matter: how hard it's raining (field strength) and how you tilt the ring (angle).

Active recall

What is magnetic flux in words?

How many magnetic field lines pierce a surface; the surface integral

\int \mathbf{B}\cdot d\mathbf{A}

Formula for flux with uniform field through a flat surface?

\Phi = BA\cos\theta

, with

\theta

between

\mathbf{B}

and the surface normal.

The angle

\theta

\Phi=BA\cos\theta

is measured between which two things?

The field

\mathbf{B}

and the surface normal

\hat{\mathbf n}

(not the surface plane).

SI unit of magnetic flux?

Weber (Wb)

= \text{T}\cdot\text{m}^2

Flux through a surface held edge-on to the field?

Zero, because

\theta=90^\circ

and

\cos90^\circ=0

(lines skim along, none pierce).

Why is the general flux written as an integral?

Field/surface may vary; slice into tiny patches where

\mathbf{B}

is constant, sum

\mathbf{B}\cdot d\mathbf{A}

, take the limit.

Total magnetic flux through any closed surface, and why?

\oint\mathbf{B}\cdot d\mathbf{A}=0

; field lines form closed loops (no monopoles), so what enters must leave.

Flux linkage for an

N

-turn coil?

N\Phi

— each turn links the flux.

Why does tilting a loop reduce its flux?

The effective (projected) area facing the field shrinks as

A\cos\theta

Connections

Faraday's Law of Induction — EMF $=-\dfrac{d\Phi}{dt}$ ; flux change drives induction.
Lenz's Law — sign of induced EMF opposes flux change.
Gauss's Law for Magnetism — $\oint\mathbf{B}\cdot d\mathbf{A}=0$ , no monopoles.
Electric Flux — analogous $\int\mathbf{E}\cdot d\mathbf{A}$ ; compare Gauss's laws.
Magnetic Field B — the field being integrated.
Inductance — defined via flux linkage per current, $L=N\Phi/I$ .
Surface and Line Integrals — the math machinery of $d\mathbf{A}$ .

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Magnetic flux ka matlab simple hai: kitni magnetic field lines kisi surface ke through nikal rahi hain. Socho rain straight neeche gir rahi hai aur tumne ek ring pakdi hai — jab ring flat (upar ki taraf face karti hui) ho to bahut saari boondein through hoti hain, jab tilt karte ho to kam, aur jab ring ko bilkul edge-on (kinare se) pakdo to ek bhi boond through nahi hoti. Yahi rain-drops ki tarah field lines hain.

Formula nikalta kaise hai? Agar field surface ko seedha (perpendicular) hit kare to $\Phi = BA$ — yaani field strength × area. Lekin jab surface tilt ho jaaye, to sirf projected area $A\cos\theta$ field ko "face" karti hai, isliye $\Phi = BA\cos\theta$ . Yahan sabse important baat: $\theta$ field aur surface ke normal (perpendicular spike) ke beech ka angle hai, surface plane ke saath wala nahi. Edge-on case mein $\theta = 90^\circ$ , $\cos 90^\circ = 0$ , isliye flux zero — bilkul rain wale logic se match karta hai.

Agar field har jagah same nahi hai ya surface curved hai, to surface ko chhote-chhote tukdo mein todo, har tukde pe $\mathbf{B}\cdot d\mathbf{A}$ nikalo aur sum (integral) karo: $\Phi = \int \mathbf{B}\cdot d\mathbf{A}$ . Yeh sirf simple case ko har point pe lagaane ka calculus version hai.

Yeh kyun important hai? Kyunki Faraday's law mein EMF tabhi banta hai jab flux change hota hai. Toh flux samajhna induction, generators, transformers — sab ki foundation hai. Aur ek bonus rule: kisi bhi closed surface ke through total flux hamesha zero hota hai, kyunki magnetic lines hamesha closed loops banati hain (no monopoles).

What is flux, exactly?

Why a dot product? (Derivation from scratch)

Gauss's law for magnetism (a free bonus)

Worked examples

Active recall

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Electromagnetism

Connections