🔄 Electromagnetic Induction & AC Circuits

Faraday’s law, motional emf, inductance, AC reactance, LR/LC transients

ε = − dΦ/dt ε = Bℓv ε = −L dI/dt ω = 1/√(LC)
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💡 Induction and AC at a glance

Changing magnetic flux Φ_B through a circuit induces emf according to Faraday’s law ε = −dΦ_B/dt; the minus sign (Lenz’s law) ensures the induced current opposes the change. Motional emf ε = Bℓv arises when a conductor moves in a field. Self-inductance L and mutual inductance M relate flux and current (ε = −L dI/dt, ε₂ = −M dI₁/dt). In AC, inductors and capacitors introduce reactances X_L = ωL and X_C = 1/(ωC), while LR/LC circuits have characteristic time and frequency scales τ_L = L/R and ω = 1/√(LC).

Induction & AC calculators

Faraday/flux, motional emf, inductance, AC reactance, LR/LC

Flux through N‑turn loop (Φ_B = B A cos θ)
Change over time (ΔB or Δθ) for induced emf
Φ_B = —, Φ_total = —, ε = —
Uses Φ_B = B A cos θ, Φ_total = N Φ_B, and ε = −N ΔΦ/Δt.
Sliding rod: ε = B ℓ v, I = ε / R
Rotating rod / disk: ε = ½ B ω L² or ½ B ω R²
ε_rod = —, I = —, ε_rot = —
Sliding rod: ε = Bℓv, I = ε/R. Rotating rod/disk: ε = ½ B ω L² or ½ B ω R².
Self‑inductance and emf (ε = −L dI/dt)
Solenoid: L = μ₀ n² A l
ε = —, U = —, L_solenoid = —
Generic: ε = −L dI/dt, U = ½ L I². Solenoid: L = μ₀ n² A l.
Mutual inductance: M = k√(L₁L₂), ε₂ = −M dI₁/dt
M = —, ε₂ = —
For two coupled coils: M = k√(L₁L₂), ε₂ = −M dI₁/dt.
AC reactance and impedance (R–L–C)
X_L = —, X_C = —, |Z_RLC| = —
Uses X_L = 2πfL, X_C = 1/(2πfC), and |Z_RLC| = √(R² + (X_L − X_C)²).
LR circuit: ε, L, R, t
LC circuit: L, C
I_LR = —, τ_L = —, ω_LC = —, T_LC = —
LR: τ_L = L/R, I(t) = (ε/R)(1 − e^{−t/τ}). LC: ω = 1/√(LC), T = 2π√(LC).

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Faraday’s law, flux & motional emf

Concept Formula Notes
Magnetic flux Φ_B = B A cos θ Wb (weber), θ angle between B and normal
Faraday’s law ε = − dΦ_B/dt, ε = −N dΦ_B/dt Induced emf; minus sign = Lenz’s law
Motional emf (rod) ε = B ℓ v v ⊥ B, ℓ length in field
Rotating rod/disk ε = ½ B ω L² (rod), ε = ½ B ω R² (disk) Between center and rim

Inductance, AC reactance, LR & LC

Concept Formula Notes
Self-inductance L = Φ/I, ε = −L dI/dt U = ½ L I²
Solenoid inductance L = μ₀ n² A l n = turns per unit length
Mutual inductance M = Φ₂₁/I₁, ε₂ = −M dI₁/dt, M = k√(L₁L₂) 0 ≤ k ≤ 1 (coupling)
AC reactance X_L = 2πfL, X_C = 1/(2πfC) Inductive: V leads I; capacitive: I leads V
RLC impedance |Z| = √(R² + (X_L − X_C)²) Series RLC
LR time constant τ_L = L / R I(t) growth: I = (ε/R)(1 − e^{−t/τ_L})
LC oscillator ω = 1/√(LC), T = 2π√(LC) Energy oscillates between L and C

About this induction & AC tool

This page brings together the core electromagnetic induction formulas needed for school physics, JEE, and NEET: Faraday’s law and flux, motional emf for rods and rotating systems, self and mutual inductance, AC reactance for R–L–C elements, and LR/LC transients. Each calculator is paired with a step-by-step derivation so you can connect the equations to the physical picture.