Self-inductance L, mutual inductance M

1.8.28 · Physics › Electromagnetism

Inductance exist kyun karta hai?

Logic ki chain, kabhi skip mat karo: $I\stackrel{\text{Ampeˋre}}{\to }B\stackrel{\text{integrate}}{\to }\Phi \stackrel{\text{Faraday}}{\to }\epsilon$

$L$ aur $M$ kya hain (flux se definitions)

Ek long solenoid ke liye $L$ derive kaise karein (scratch se)

Ek solenoid: length $\ell$, $N$ turns, cross-section $A$, $n=N/\ell$ turns per metre.

Step 1 — Andar field (Ampère's law). $B={\mu }_{0}nI={\mu }_0\frac{N}{\ell }I$ Yeh step kyun? Ek long solenoid ke andar field uniform aur axial hota hai; ek Amperian rectangle deta hai $B{\ell }_{\text{path}}={\mu }_0(n{\ell }_{\text{path}})I$.

Step 2 — EK turn se Flux. $\Phi =BA={\mu }_0\frac{N}{\ell }IA$ Yeh step kyun? Flux $=\int \vec{B}\cdot d\vec{A}=BA$ kyunki $B$ uniform hai aur cross-section ke perpendicular hai.

Step 3 — Flux LINKAGE (saare $N$ turns). $N\Phi ={\mu }_0\frac{N^2}{\ell }AI$ Yeh step kyun? $N$ turns mein se har ek same $\Phi$ se thread hota hai, isliye total linkage $N\Phi$ hai.

Step 4 — $L=N\Phi /I$ read off karo. $\boxed{L={\mu }_0\frac{N^{2}A}{\ell }={\mu }_0{n}^{2}A\ell }$ Yeh kyun important hai: $L\propto N^2$ — turns double karne se inductance chaar guna ho jaata hai, kyunki har turn dono zyada flux banata hai aur zyada flux link karta hai.

Do coaxial solenoids ke liye $M$ derive kaise karein

Andar wala long solenoid (source) $n_1=N_1/\ell$ ke saath; bahar/overlying coil $N_2$ turns wala same area $A$, length $\ell$ par.

Step 1. Inner current $I_1$ andar $B_1={\mu }_0{n}_1{I}_1$ banata hai. Step 2. Coil 2 ke ek turn se flux: ${\Phi }_{21}=B_{1}A={\mu }_0{n}_1{I}_{1}A$ (sirf inner-solenoid area field carry karta hai). Step 3. Coil 2 mein linkage: $N_2{\Phi }_{21}={\mu }_0{N}_2{n}_{1}A{I}_1$. Step 4. $\boxed{M=\frac{N_2{\Phi }_{21}}{I_1}={\mu }_0\frac{N_1{N}_2}{\ell }A={\mu }_0{n}_1{N}_{2}A}$

Stored energy (kyun ek inductor current ke liye flywheel hai)

Worked examples

Common mistakes (steel-manned)

Active-recall flashcards

Connections

• Faraday's law of induction — parent law $\epsilon =-d(N\Phi )/dt$.
• Lenz's law — minus sign explain karta hai.
• Ampère's law — solenoid ke andar $B$ deta hai.
• Magnetic flux — woh quantity jo link ho rahi hai.
• Energy stored in magnetic field — $U=\frac{1}{2}L{I}^2$ aur $u=B^2/2{\mu }_0$.
• Transformers — $M$ aur $k\approx 1$ ka application.
• RL circuits — jahan $LdI/dt$ current growth/decay ko shape karta hai.

Concept Map