Question bank — Magnetic flux Φ = ∫B·dA

True. Magnetic field lines closed loops banati hain jinka koi start ya end nahi hota (koi monopole nahi), isliye $\oint \mathbf{B}\cdot d\mathbf{A}=0$ kisi bhi closed surface ke liye.

False. Agar loop edge-on ho ($\theta =90^{\circ }$), toh $\cos \theta =0$ aur flux zero hoga chahe $\mathbf{B}$ kitna bhi strong ho — orientation utna hi matter karta hai jitna strength.

False. Dot product dono vectors ko ek single number mein collapse kar deta hai, isliye $\Phi$ ek scalar hai (iska magnitude aur sign hai, lekin koi direction nahi).

False. Ye $\mathbf{B}$ aur surface normal $\hat{\mathbf{n}}$ ke beech ka angle hai. "Normal's the boss."

False. Har single turn ke through flux $\Phi$ unchanged rehta hai; jo double hota hai wo hai flux linkage $N\Phi$, jo Faraday's EMF mein appear karta hai.

True. Agar $\theta >90^{\circ }$ ho toh field surface ke "back" se enter karta hai, $\cos \theta <0$, isliye $\Phi <0$ — sign ye encode karta hai ki lines chosen normal ke relative kis taraf pierce karti hain.

True. $\hat{\mathbf{n}}$ flip karne se $\theta$ ki jagah $180^{\circ }-\theta$ aa jaata hai, isliye $\cos \theta \to -\cos \theta$; flux ka magnitude ek physical fact hai, sign ek bookkeeping convention hai.

True. Sirf area ka woh projection jo $\mathbf{B}$ ke perpendicular plane par hai, lines ko pakadta hai; $\Phi =B(A\cos \theta )$.

Error: "field ke perpendicular" loop ke plane ko describe karta hai; tab normal $\mathbf{B}$ ke parallel hota hai, isliye $\theta =0^{\circ }$ aur $\Phi =BA$ (maximum) hoga, zero nahi.

Error: $\Phi =BA$ tab hi hold karta hai jab field flat surface ke bhi perpendicular ho. Generally phir bhi $BA\cos \theta$ chahiye, aur curved surface ya non-uniform field integral force karta hai.

Error: koi magnetic charge nahi hota. Magnetic Gauss's law hamesha $\oint \mathbf{B}\cdot d\mathbf{A}=0$ hai — right side hamesha zero fixed hai, electric case ke unlike.

Error: plane mein padi lines surface ke saath skim karti hain aur kuch bhi pierce nahi karti. Yahan $\theta =90^{\circ }$ hai (normal $\perp \mathbf{B}$), jo $\Phi =0$ deta hai.

Error: flux field times area hai, isliye iska unit weber hai ($1\text{ Wb}=1\text{ T}\cdot {\text{m}}^2$). Tesla sirf $\mathbf{B}$ ka unit hai.

Error (avoidable work): curved cap ke through flux uske flat circular rim ke through flux ke barabar hai, kyunki uniform field mein $\oint \mathbf{B}\cdot d\mathbf{A}=0$ closed surface (cap + disc) par hold karta hai. Aasaan flat disc use karo.

Error: sirf tab jab dono same field orientation catch karein. Edge-on rakha bada loop zero flux rakhta hai jabki face-on chota loop positive flux rakhta hai.

Kyunki sirf field ka woh hissa jo normal ke along hai, surface ko pierce karta hai; dot product $\mathbf{B}\cdot d\mathbf{A}=BA\cos \theta$ automatically woh perpendicular component extract kar leta hai aur sliding-along part discard kar deta hai.

Kyunki magnetic field lines unbroken closed loops hoti hain — jo line andar jaati hai woh bahar bhi aati hai. Koi monopole andar nahi hota jo source ya sink ka kaam kare, isliye ins aur outs exactly cancel ho jaate hain.

Taaki har patch par field essentially constant ho aur simple rule $\mathbf{B}\cdot d\mathbf{A}$ apply ho sake; sum karna aur limit lena (integral) varying fields ya curved surfaces ke liye sach mein total rebuild karta hai.

Kyunki induced EMF respond karta hai is change ko ki kitni lines loop ko thread karti hain — woh count flux hai. Changing $\mathbf{B}$, changing area, ya changing tilt teeno use drive kar sakte hain, aur flux teeno ko package karta hai.

Normal woh ek direction hai jo measure karta hai ki patch field ke kitna "face-on" hai; field ko uss par project karna true line-piercing contribution deta hai, jo exactly wahi hai jo flux ko count karna chahiye.

Tilt karne se projected area $A\cos \theta$ jo field ki taraf face karta hai woh shrink ho jaata hai, isliye kam lines pass through hoti hain; $\theta =90^{\circ }$ par projection vanish ho jaata hai aur flux zero ho jaata hai.

Zero. $\cos 90^{\circ }=0$, isliye $\Phi =BA\cos 90^{\circ }=0$ — lines surface ke parallel chalti hain aur koi bhi ise pierce nahi karta.

Zero. $\mathbf{B}=0$ ke saath, integrand $\mathbf{B}\cdot d\mathbf{A}=0$ orientation chahe kuch bhi ho har patch par, isliye $\Phi =0$.

Zero. $dA$ ka sum kuch bhi nahi hota isliye lines cross karne ke liye koi area nahi hai; $\Phi =0$ trivially.

$\cos \theta$ thoda negative ho jaata hai, isliye $\Phi$ smoothly positive se negative mein cross karta hai jab loop edge-on se aage tip karta hai — field ab surface ke chosen "back" se enter karta hai.

Exactly zero. Uniform field mein har line jo ek side se enter karti hai doosri side se exit karti hai; aur zyada fundamentally $\oint \mathbf{B}\cdot d\mathbf{A}=0$ sabhi closed surfaces ke liye hold karta hai.

Pura maximum, $100\%$. Yahan $\theta =0$, $\cos 0=1$, isliye $\Phi =BA$ — har line head-on pierce karti hai.

Ye maximum $BA$ se shuru hota hai (wahan change ki rate zero hai) aur edge-on par $0$ tak girta hai; kyunki $\Phi \propto \cos \theta$, flux $\theta =90^{\circ }$ ke paas sabse fast change karta hai aur $\theta =0^{\circ }$ ke paas sabse slow.