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CLASS 12 • PHYSICS
Electromagnetic Induction
1. Flux & Faraday's Law
Magnetic Flux
$\Phi_B = \vec{B} \cdot \vec{A} = BA \cos\theta$
$\theta$ = Angle with Normal
Faraday's Law
$\varepsilon = -N \frac{d\Phi_B}{dt}$
Magnitude $|\varepsilon| = N \frac{\Delta \Phi}{\Delta t}$
Induced Current
$I = \frac{|\varepsilon|}{R}$
Induced Charge
$q = \frac{N}{R} (\Delta \Phi)$
Independent of Time!
2. Motional EMF
Rod Moving in Field
$\varepsilon = Bvl$
Only if B, v, l are mutually $\perp$
Rod Rotating in Field
$\varepsilon = \frac{1}{2} B \omega l^2 = B f (\pi l^2)$
Between Center and Rim
Force Required
$F = IlB = \frac{B^2 l^2 v}{R}$
Power Dissipated
$P = Fv = \frac{B^2 l^2 v^2}{R}$
3. Self Induction (L)
Basic Relations
$\Phi = LI$ and $\varepsilon = -L \frac{dI}{dt}$
Long Solenoid
$L = \frac{\mu_0 N^2 A}{l} = \mu_0 n^2 Al$
$n = N/l$ (Turns density)
Energy Stored
$U = \frac{1}{2} L I^2$
Magnetic Potential Energy
Energy Density
$u_B = \frac{B^2}{2\mu_0}$
4. Mutual Induction (M)
Basic Relations
$\Phi_2 = M I_1$ and $\varepsilon_2 = -M \frac{dI_1}{dt}$
Two Long Solenoids
$M = \frac{\mu_0 N_1 N_2 A}{l} = \mu_0 n_1 n_2 Al$
Coupling Coefficient
$k = \frac{M}{\sqrt{L_1 L_2}}$
Max value k=1 (Tight coupling)
5. AC Generator
Instantaneous Flux
$\Phi = NBA \cos(\omega t)$
Induced EMF
$\varepsilon = \varepsilon_0 \sin(\omega t)$
Peak EMF ($\varepsilon_0$)
$\varepsilon_0 = NBA\omega$
Also called Voltage Amplitude
6. Inductors Combination
Series Combination
$L_{eq} = L_1 + L_2$
(Assuming Mutual Inductance M=0)
Parallel Combination
$\frac{1}{L_{eq}} = \frac{1}{L_1} + \frac{1}{L_2}$
With Mutual (Series Helping)
$L = L_1 + L_2 + 2M$
With Mutual (Series Opposing)
$L = L_1 + L_2 - 2M$
7. Common Exam Traps
- ⚠️ Flux $\neq$ Field: Flux can be zero even if B is non-zero (if $\theta=90^\circ$). EMF depends on CHANGE in flux, not flux itself.
- ⚠️ Induced Charge: It depends ONLY on the change in flux ($\Delta \Phi$). It does NOT depend on how fast or slow the change happens (Time).
- ⚠️ Lenz's Law: The induced current opposes the CAUSE. If magnet approaches, face becomes same pole (Repulsion). If magnet leaves, face becomes opposite pole (Attraction).
8. Golden Theory Rules
- Eddy Currents: Circulating currents in bulk conductors. Minimized by using LAMINATED cores. Used in Magnetic Brakes.
- Self Induction: Often called "Electrical Inertia" because it opposes any change in current.
- Solenoid with Core: Inserting a soft iron core drastically INCREASES 'L' (by factor $\mu_r$) and hence increases induced EMF.
