Self-inductance L, mutual inductance M

WHY does inductance exist at all?

The chain of logic, never skipped: $I \;\xrightarrow{\text{Ampère}}\; B \;\xrightarrow{\text{integrate}}\; \Phi \;\xrightarrow{\text{Faraday}}\; \varepsilon$

WHAT are $L$ and $M$ (definitions from flux)

HOW to derive $L$ for a long solenoid (from scratch)

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

Step 1 — Field inside (Ampère's law). $B = \mu_0 n I = \mu_0\frac{N}{\ell}I$ Why this step? Inside a long solenoid the field is uniform and axial; an Amperian rectangle gives $B\ell_{\text{path}} = \mu_0 (n\ell_{\text{path}})I$ .

Step 2 — Flux through ONE turn. $\Phi = B A = \mu_0\frac{N}{\ell}I\,A$ Why this step? Flux $=\int \vec B\cdot d\vec A = BA$ because $B$ is uniform and perpendicular to the cross-section.

Step 3 — Flux LINKAGE (all $N$ turns). $N\Phi = \mu_0\frac{N^2}{\ell}A\,I$ Why this step? Each of the $N$ turns is threaded by the same $\Phi$ , so total linkage is $N\Phi$ .

Step 4 — Read off $L = N\Phi/I$ . $\boxed{\,L = \mu_0\frac{N^2 A}{\ell} = \mu_0 n^2 A\,\ell\,}$ Why this matters: $L\propto N^2$ — doubling turns quadruples inductance, because each turn both makes more flux and links more flux.

HOW to derive $M$ for two coaxial solenoids

Inner long solenoid (the source) with $n_1=N_1/\ell$ ; outer/overlying coil of $N_2$ turns on the same area $A$ , length $\ell$ .

Step 1. Inner current $I_1$ makes $B_1=\mu_0 n_1 I_1$ inside. Step 2. Flux through one turn of coil 2: $\Phi_{21}=B_1 A=\mu_0 n_1 I_1 A$ (only the inner-solenoid area carries field). Step 3. Linkage in coil 2: $N_2\Phi_{21}=\mu_0 N_2 n_1 A\,I_1$ . Step 4. $\boxed{\,M = \frac{N_2\Phi_{21}}{I_1}=\mu_0\frac{N_1 N_2}{\ell}A = \mu_0 n_1 N_2 A\,}$

Figure — Self-inductance L, mutual inductance M

Energy stored (why an inductor is a flywheel for current)

Worked examples

Common mistakes (steel-manned)

Recall Feynman: explain to a 12-year-old

Imagine pushing a heavy swing. A coil of wire is like a swing for electric current: once the current is moving, it doesn't want to suddenly stop or change — it "swings back" with a little electrical push (the back-EMF). Self-inductance is how stubborn the swing is about its own motion. Mutual inductance is when your swing is tied by a rope to a friend's swing: when you change yours, theirs feels a tug too. The more loops in the wire and the tighter they're packed, the more stubborn the swing.

Active-recall flashcards

What is the defining relation for self-inductance from flux?

N\Phi = LI

, so

L=N\Phi/I

(flux linkage per amp), unit henry.

Self-induced EMF formula and its sign reason?

\varepsilon=-L\,dI/dt

; minus from Lenz's law (opposes the change).

Why does a steady (constant) current produce no self-EMF?

Because

dI/dt=0

; EMF depends on rate of change of current, not its value.

Inductance of a long solenoid?

L=\mu_0 N^2 A/\ell=\mu_0 n^2 A\ell

Why is

L\propto N^2

Each turn both produces extra flux (

\propto N

) and links the flux (

\times N

), compounding to

N^2

Mutual inductance definition?

M=N_2\Phi_{21}/I_1

; EMF in coil 2 is

\varepsilon_2=-M\,dI_1/dt

State the reciprocity result for

M

M_{12}=M_{21}=M

always, regardless of coil sizes.

Coupling coefficient relation?

M=k\sqrt{L_1L_2}

with

0\le k\le1

;

k=1

= perfect flux linkage.

Energy stored in an inductor and its derivation idea?

U=\tfrac12 LI^2

, from

\int LI\,dI

= work done against back-EMF.

SI unit of inductance in base/derived terms?

Henry

=

Wb/A

=

V·s/A.

Connections

Faraday's law of induction — the parent law $\varepsilon=-d(N\Phi)/dt$ .
Lenz's law — explains the minus sign.
Ampère's law — gives $B$ inside the solenoid.
Magnetic flux — the quantity being linked.
Energy stored in magnetic field — $U=\tfrac12 LI^2$ and $u=B^2/2\mu_0$ .
Transformers — application of $M$ and $k\approx1$ .
RL circuits — where $L\,dI/dt$ shapes current growth/decay.

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, self-inductance $L$ ka matlab simple hai: jab kisi coil mein current behta hai, woh apna magnetic field banata hai aur flux coil ke through pass hota hai. Ab agar current change hota hai, to flux change hota hai, aur Faraday's law kehta hai ki badalta flux EMF paida karta hai. Yeh EMF change ko oppose karta hai (Lenz's law, isliye minus sign): $\varepsilon=-L\,dI/dt$ . Isliye coil ek "current ka flywheel" jaisa behave karti hai — current ko achanak badalne nahi deti. Yaad rakho: EMF current par nahi, current ke rate of change par depend karta hai. Constant current = zero self-EMF.

Solenoid ke liye derive karna easy hai: andar field $B=\mu_0 nI$ , ek turn ka flux $\Phi=BA$ , aur total linkage $N\Phi$ , to $L=\mu_0 N^2 A/\ell$ . Yahan dhyan do — $L$ , $N^2$ ke proportional hai, sirf $N$ ke nahi, kyunki har turn flux banata bhi hai aur link bhi karta hai (double duty). Yeh ek classic mistake hai jo students karte hain.

Mutual inductance $M$ tab aata hai jab do coils paas-paas hoti hain. Coil 1 ka current coil 2 mein flux bhejta hai; agar $I_1$ change ho to coil 2 mein EMF aata hai: $\varepsilon_2=-M\,dI_1/dt$ . Ek important fact: $M_{12}=M_{21}$ , chahe coils chhoti-badi kuch bhi ho (reciprocity). Aur $M=k\sqrt{L_1L_2}$ , jahan $k$ coupling batata hai — $k=1$ matlab perfect coupling, jaise ideal transformer mein.

Practical importance: transformers, ignition coils, induction cooktops, aur RL circuits sab inhi ideas par chalte hain. Exam mein zyada marks isi se aate hain: $L=\mu_0 N^2A/\ell$ , $\varepsilon=-L\,dI/dt$ , $U=\tfrac12 LI^2$ , aur $M=k\sqrt{L_1L_2}$ — yeh chaar formule pakke kar lo, baaki sab inhi se nikal aata hai.

WHY does inductance exist at all?

WHAT are $L$ and $M$ (definitions from flux)

HOW to derive $L$ for a long solenoid (from scratch)

HOW to derive $M$ for two coaxial solenoids

Energy stored (why an inductor is a flywheel for current)

Worked examples

Common mistakes (steel-manned)

Active-recall flashcards

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Electromagnetism

Connections