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Electromagnetic Induction (EMI) Formulas

Faraday's law, Lenz's law, motional EMF, self & mutual inductance, and LR growth/decay circuits.

9 Formulas•490+ Questions in Bank

Core Formulas & Mathematical Identities

9
01

Faraday's & Lenz's Law

ε=−dΦBdt,ΦB=B⃗⋅A⃗=BAcos⁡θ\varepsilon = -\frac{d\Phi_B}{dt}, \quad \Phi_B = \vec{B} \cdot \vec{A} = B A \cos\theta

Induced electromotive force opposing changing magnetic flux.

Examiner Tip: Total induced charge q = ΔΦ / R is independent of the rate of change of flux.
02

Motional EMF

ε=(v⃗×B⃗)⋅l⃗=Bvl  (Translational),ε=12BωL2  (Rotating Rod)\varepsilon = (\vec{v} \times \vec{B}) \cdot \vec{l} = B v l \; (\text{Translational}), \quad \varepsilon = \frac{1}{2}B \omega L^2 \; (\text{Rotating Rod})

Voltage induced across conductors moving through magnetic field.

Examiner Tip: Induced current direction given by Fleming's Right-Hand Rule.
03

Self & Mutual Inductance

Φ=LI  ⟹  ε=−LdIdt,Lsolenoid=μ0n2Al=μ0N2Al,M=kL1L2\Phi = L I \implies \varepsilon = -L \frac{dI}{dt}, \quad L_{\text{solenoid}} = \mu_0 n^2 A l = \frac{\mu_0 N^2 A}{l}, \quad M = k\sqrt{L_1 L_2}

Inertia of electric current in coils and coupling coefficient k.

Examiner Tip: Stored magnetic energy density: u_B = B² / (2μ₀).
04

Transient LR Circuit (Current Growth & Decay)

I(t)=VR(1−e−t/τ)  (Growth),I(t)=I0e−t/τ  (Decay),τ=LRI(t) = \frac{V}{R}(1 - e^{-t / \tau}) \; (\text{Growth}), \quad I(t) = I_0 e^{-t / \tau} \; (\text{Decay}), \quad \tau = \frac{L}{R}

Inductive time constant governing current evolution.

Examiner Tip: At t = 0, inductor acts as open circuit; at t -> ∞, acts as ideal short wire.
05

Non-Conservative Induced Electric Field

∮E⃗⋅dl⃗=−dΦBdt  ⟹  E(r)={r2dBdt(r≤R)R22rdBdt(r>R)\oint \vec{E} \cdot d\vec{l} = -\frac{d\Phi_B}{dt} \implies E(r) = \begin{cases} \frac{r}{2}\frac{dB}{dt} & (r \le R) \\ \frac{R^2}{2r}\frac{dB}{dt} & (r > R) \end{cases}

Electric field produced by time-varying magnetic flux inside and outside a cylindrical field region of radius R.

Examiner Tip: Unlike electrostatic fields, induced electric field lines form closed continuous loops and work done along a closed path is non-zero.
06

Magnetic Force & Power Dissipated in Moving Loop

Fmagnetic=B2L2vR,Pmechanical=PJoule=B2L2v2R,v(t)=v0e−t/τF_{\text{magnetic}} = \frac{B^2 L^2 v}{R}, \quad P_{\text{mechanical}} = P_{\text{Joule}} = \frac{B^2 L^2 v^2}{R}, \quad v(t) = v_0 e^{-t / \tau}

Retarding Lorentz braking force on a conducting rod sliding across uniform field rails with load resistance R.

Examiner Tip: Mechanical power input equals rate of Joule heat dissipation in the circuit, verifying energy conservation.
07

Inductance Combinations & Stored Magnetic Energy

UB=12LI2,uB=B22μ0,Lseries=L1+L2±2M,Lparallel=L1L2−M2L1+L2∓2MU_B = \frac{1}{2} L I^2, \quad u_B = \frac{B^2}{2\mu_0}, \quad L_{\text{series}} = L_1 + L_2 \pm 2M, \quad L_{\text{parallel}} = \frac{L_1 L_2 - M^2}{L_1 + L_2 \mp 2M}

Total magnetic energy stored in current-carrying inductor and equivalent inductance of coupled coils.

Examiner Tip: Use +2M when magnetic fluxes aid each other; use -2M when fluxes oppose each other.
08

AC Generator (Rotating Coil) EMF

ε=NBAωsin⁡(ωt)=ε0sin⁡(ωt),ε0=NBAω,εrms=ε02=NBAω2\varepsilon = NBA\omega\sin(\omega t) = \varepsilon_0\sin(\omega t), \quad \varepsilon_0 = NBA\omega, \quad \varepsilon_{\text{rms}} = \frac{\varepsilon_0}{\sqrt{2}} = \frac{NBA\omega}{\sqrt{2}}

EMF produced by N-turn coil of area A rotating with angular velocity ω in uniform magnetic field B. Peak EMF ε₀ = NBAω. Coil plane is parallel to B at maximum EMF; perpendicular to B at zero EMF.

Examiner Tip: Maximum EMF occurs when coil plane is parallel to B (flux = 0, rate of change of flux is maximum). Zero EMF when coil is perpendicular to B (flux maximum, rate of change = 0). Phase difference between flux and EMF is 90°.
09

Eddy Currents — Power Loss & Applications

Peddy∝B2f2t2/ρresistivity,Skin depth: δ=2ρωμP_{\text{eddy}} \propto B^2 f^2 t^2 / \rho_{\text{resistivity}}, \quad \text{Skin depth: } \delta = \sqrt{\frac{2\rho}{\omega \mu}}

Eddy currents are induced in bulk conductors by changing B. Loss is reduced by laminating the core (thin silicon steel sheets increase resistivity path). Power loss ∝ (frequency)² × (thickness)² × (B)².

Examiner Tip: Applications: electromagnetic braking (train brakes), induction furnace (heating by eddy currents), galvanometer damping, speedometers. Silicon steel (high resistivity) is used in transformer cores to minimise eddy current losses.

Top Examiner Traps & Pitfalls

Frequently tested misconceptions in Electromagnetic Induction (EMI)

  • •Minus sign in Faraday's law represents Lenz's law (induced EMF opposes the CHANGE in magnetic flux).
  • •Self-inductance of solenoid depends on square of turns: L = μ₀ N² A / l.
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