Magnetic field of straight wire, circular loop, solenoid, toroid

1.8.24 · Physics › Electromagnetism

Do master laws (YE KYUN EXIST KARTE HAIN)

YE FORM KYUN? Yeh Coulomb ke $1/r^2$ law ka magnetic analogue hai, lekin cross product $d\vec{l}\times \hat{r}$ yeh encode karta hai ki magnetic field current se door point karne ki jagah current ke around circulate karta hai.

KAUN SA KAB USE KAREIN? Ampère's law tab use karo jab symmetry tumhe $B$ ko integral se bahar nikalne de (straight wire, solenoid, toroid). Biot–Savart tab use karo jab koi helpful symmetry na ho (loop ke axis par point).

1. Straight Wire — Ampère's Law (KAISE)

Scratch se Derivation: $\oint \vec{B}\cdot d\vec{l}=B\oint dl=B(2\pi r)$ Yeh step kyun? $B$ circle par magnitude mein constant hai aur $d\vec{l}$ ke parallel hai har jagah, toh dot product sirf $Bdl$ hai.

${\mu }_0{I}_{\text{enc}}={\mu }_{0}I$ ke barabar set karo: $B(2\pi r)={\mu }_{0}I\Rightarrow \boxed{B=\frac{{\mu }_{0}I}{2\pi r}}$

Finite wire (Biot–Savart result) — ek segment ke liye jo point par angles ${\theta }_1,{\theta }_2$ subtend karta hai: $B=\frac{{\mu }_{0}I}{4\pi r}(\sin {\theta }_1+\sin {\theta }_2)$ Infinite wire ke liye ${\theta }_1={\theta }_2=90°$, jo wapas $\frac{{\mu }_{0}I}{2\pi r}$ deta hai. ✓

2. Circular Loop (axis par) — Biot–Savart (KAISE)

Koi simple Amperian loop kaam nahi karta, toh integrate karo.

Derivation: Har element: $d\vec{l}\perp \hat{r}$, toh $|d\vec{l}\times \hat{r}|=dl$. Distance $r=\sqrt{a^2+x^2}$. $dB=\frac{{\mu }_{0}Idl}{4\pi (a^2+x^2)}$ Axial part: $\cos \alpha =\frac{a}{\sqrt{a^2+x^2}}$ se multiply karo. $B=\int dB\cos \alpha =\frac{{\mu }_{0}I}{4\pi }\frac{a}{(a^2+x^2{)}^{3/2}}\oint dl$ Yeh step kyun? $\oint dl=2\pi a$, circumference hai. $\boxed{B=\frac{{\mu }_{0}I{a}^2}{2(a^2+x^2{)}^{3/2}}}$

3. Solenoid — Ampère's Law (KAISE)

Derivation: Length $L$ ka ek rectangular Amperian loop lo, ek side andar (field $B$, parallel), opposite side bahut bahar (field ≈ 0), do short sides perpendicular (contribute 0). $\oint \vec{B}\cdot d\vec{l}=BL$ Enclosed current = (turns in length $L$) × $I=nL\cdot I$, jahan $n=N/L$ turns per metre hai. $BL={\mu }_0(nL)I\Rightarrow \boxed{B={\mu }_{0}nI}$

4. Toroid — Ampère's Law (KAISE)

Derivation: Amperian loop = core ke andar radius $r$ ka circle. Symmetry se $B$ us par constant hai: $\oint \vec{B}\cdot d\vec{l}=B(2\pi r)$ Enclosed current = $NI$ (loop saare $N$ turns ko ek baar cross karta hai). $B(2\pi r)={\mu }_{0}NI\Rightarrow \boxed{B=\frac{{\mu }_{0}NI}{2\pi r}}$

Forecast-then-Verify checkpoint

Answers padhne se pehle, predict karo:

1. Wire mein current double karo → field? (double ho jaayegi)
2. Wire se do guna door jao → field? (aadhi ho jaayegi)
3. Solenoid ko same total turns ke saath do guna lamba kheencho → andar field? (aadhi ho jaayegi, kyunki $n$ aadha ho jaata hai)
4. Toroid field inner vs outer edge par → kaun sa bada? (inner, kyunki $B\propto 1/r$)

Flashcards

Connections

• Biot-Savart Law — loop aur finite-wire fields ke peeche integration engine.
• Ampere's Law — wire, solenoid, toroid ke liye symmetry shortcut.
• Magnetic force on a current-carrying conductor — yeh fields forces exert karte hain.
• Bar magnet and magnetic dipole — loop/solenoid dipoles ki tarah behave karte hain.
• Faraday's law of induction — changing solenoid field EMF induce karta hai.
• Magnetic flux — uniform solenoid field flux easy banata hai: $\Phi =BA$.

Concept Map