Question bank — Magnetic field of straight wire, circular loop, solenoid, toroid

True — $B={\mu }_{0}I/2\pi r$ blow up karta hai jab $r\to 0$, to $r$ ko chhota karne ya $I$ badhane par ek idealised line wire ke paas arbitrarily large field milta hai.

False — metal ke andar, axis se $r$ radius par ek Amperian circle draw karo; wo sirf uss fraction of current ko enclose karta hai jo usme se thread karta hai; woh fraction hai $I{r}^2/R^2$, jisse milta hai $B={\mu }_{0}Ir/2\pi R^2$, jo centre par zero se seedha upar grow karta hai surface par peak tak (figure dekho).

False — ye sirf total circulation fix karta hai ($\vec{B}$ loop ke tangent ka running sum), har point par $\vec{B}$ nahi. Genuine counterexample: ek solenoid ke bahar, ek Amperian loop zero net current enclose kar sakta hai phir bhi ek aisi region mein ho jahan thoda stray field abhi bhi exist karta hai — $\vec{B}$ ke plus aur minus contributions loop ke around sirf zero sum tak cancel hote hain jabki $\vec{B}$ khud nonzero hai.

True — aur yahan kyun hai: har turn apna khud ka chhota push of field tube ke andar add karta hai, aur pushes stack hote hain. Usi length mein double turns matlab double stacked pushes per metre, to andar ki steady breeze double strong hoti hai. Symbolically $n=N/L$ double hota hai, aur $B={\mu }_{0}nI$ follow karta hai.

False — loop ka field sirf uski axis par cleanly known hai; off-axis aur plane mein ye complicated aur non-uniform hai, wire ke bilkul paas sabse strong.

True (ideally) — ek perfectly wound toroid ke liye $B=0$ central hole mein aur bahar; sirf finite winding pitch ki chhoti leakage escape karti hai.

True — $\vec{B}$ yahan har formula mein $I$ mein linear hai, to current ka sign flip karne par right-hand rule se $\vec{B}$ flip ho jaata hai.

False — dono fields midpoint par same direction mein point karte hain (grip rule), to wo add hote hain, cancel nahi; wo far outside cancel hote hain, beech mein nahi.

Error — axis ke around koi bhi Amperian path constant $B$ parallel to $d\vec{l}$ nahi rakhta, to $B$ bahar nahi nikal sakta; tumhe Biot-Savart Law ke saath integrate karna hoga.

Error — $n=N/L$ mein koi radius nahi hai; ideal long-solenoid field genuinely radius-independent hai, to ise fatter banana kuch nahi badlta.

Error — $r$ doughnut ke centre se field point tak ki doori hai (core ke andar), wire ki thickness ya core ka minor radius nahi.

Error — mouth par field exactly $\frac{1}{2}{\mu }_{0}nI$ tak drop ho jaata hai. Kyun: andar deeply, ek coil ka slice apne dono sides par wire rakhta hai jo field uske andar feed karta hai; bilkul end par, sirf aadha coil bachta hai (sab ek side par), to wo aadha push deliver karta hai. Field lines wahan fan out karke weak ho jaati hain — figure mein flaring lines dekho.

Error — infinite wire ke liye ends $\pm 90^{\circ }$ tak jaate hain, to ${\theta }_1={\theta }_2=90^{\circ }$ aur $\sin =1$, jisse ${\mu }_{0}I/2\pi r$ recover hota hai; inhe $0$ set karne par galat tarike se zero aayega.

Error — $N$ bhool gaye; har turn contribute karta hai, to centre field $N{\mu }_{0}I/2a$ hai.

Error — element law $1/r^2$ hai, lekin poori infinite wire integrate karne ke baad result $1/r$ hai; sirf point-charge (ya point element) case $1/r^2$ hai.

Wire ke liye Amperian path ek circle hai circumference $2\pi r$ ke saath, aur $B$ us poori length se divide hota hai. Loop centre ke liye tum instead $dB={\mu }_{0}Idl/4\pi a^2$ ko ring ke har element se add karte ho; ring par $dl$ sum karne par $2\pi a$ milta hai, aur $\frac{2\pi a}{4\pi a^2}=\frac{1}{2a}$ — $\pi$ cancel ho jaata hai, clean ${\mu }_{0}I/2a$ bachta hai. Figure mein ring geometry dekho.

Ring ke bilkul opposite ends par do current elements lo. Har ek ek $d\vec{B}$ produce karta hai jo partly axis ke saath aur partly sideways (axis ki taraf) jhukta hai. Unke sideways parts bilkul opposite directions mein point karte hain aur size mein equal hain, to wo pair by pair cancel ho jaate hain; sirf axial parts, jo same direction mein point karte hain, survive karke add hote hain. Figure mein ek aisa opposite pair dikhta hai sideways arrows ke saath jo annihilate ho rahe hain.

Saare loops se return field ek huge region par spread hota hai, aur bahar ek Amperian loop zero net current enclose karta hai, to circulation vanish ho jaata hai; symmetry ke saath mila kar ye $B\approx 0$ force karta hai.

Uska Amperian circle circumference $2\pi r$ rakhta hai jo core mein vary karta hai, to $B\propto 1/r$ — inner edge par bada (chhota $r$), outer edge par chhota; sirf ek thin core uniform dikhta hai.

Magnetic field current ke around circulate karta hai nah ki usse door point karta, aur cross product exactly woh operation hai jo "flow ki direction" plus "point ki taraf direction" ko us perpendicular, circulating direction mein turn karta hai.

Ek finite wire mein poori translational symmetry nahi hoti, to $B$ kisi bhi circle ke around constant nahi hota aur integral se bahar nahi nikal sakta — tumhe $\sin {\theta }_1+\sin {\theta }_2$ result ke saath Biot-Savart Law par wapas jaana padta hai.

Uska field hole se thread karta hai aur bahar exactly ek dipole ki tarah loop back karta hai, use ek north aur south face deta hai — Bar magnet and magnetic dipole ki basis.

Idealised line wire ke liye undefined/singular ($B\to \infty$); finite radius $R$ ki real wire ke liye, $B\to 0$ true centre par jahan enclosed current zero hai.

Zero — hole mein ek Amperian circle koi current enclose nahi karta, to $\oint \vec{B}\cdot d\vec{l}=0$ aur symmetry se $B=0$ wahan.

$B\to {\mu }_{0}I{a}^2/2x^3$, $1/x^3$ dipole falloff — loop door se ek point magnetic dipole ki tarah dikhta hai (Bar magnet and magnetic dipole dekho).

Interior field ${\mu }_{0}nI\to 0$; same turns ko infinite length par spread karna unhe kuch nahi tak dilute kar deta hai.

Har formula $B=0$ deta hai — koi moving charge nahi, koi magnetic field nahi, chahe wire kitni bhi cleverly mori ho.

$B={\mu }_{0}nI$ approximation break down ho jaati hai; ek turn ke saath tumhare paas ek single loop hai aur loop formula ${\mu }_{0}I/2a$ use karna padega, long-solenoid result nahi.

Axial formula apply nahi hota; wahan field genuinely off-axis hai, interior direction ke opposite wapas curl karta hai, aur poore Biot-Savart Law integration ki zaroorat hai — koi simple closed form nahi.