1
Magnetic Flux and Faraday's Law
Magnetic Flux
- Φ = B·A·cosθ (θ = angle between B and area vector A)
- Scalar; SI unit = Weber (Wb) = T·m²; CGS unit = Maxwell; 1 Wb = 10⁸ Maxwell
- Dimension: [ML²T⁻²A⁻¹]
- Angle between B and area vector A is always taken as 90° from surface plane
Faraday's Law: EMF = −dΦ/dt (induced EMF = rate of change of flux)
Average EMF = −ΔΦ/Δt | Induced current I = EMF/R = −(1/R) × dΦ/dt
Based on Conservation of Energy
Lenz's Law
- Direction of induced current opposes the cause that produces it
- MR Law: Agar front se Anticlockwise then Picche se Clockwise (and vice versa)
- Flux increasing → induced current creates B opposite to external B (opposes increase)
- Flux decreasing → induced current creates B in same direction as external B (opposes decrease)
- Coil acts like inertia — opposes change in flux
- Lenz's Law is consequence of Conservation of Energy
EMF for rod of length L moving with velocity v in field B: EMF = BLv sinθ
For v ⊥ B ⊥ L: EMF = BLv (maximum)
Induced current: I = BLv/R
Force on rod: F = BIL = B²L²v/R (opposing motion)
Power dissipated: P = F·v = B²L²v²/R
Rotating Rod/Disc in Magnetic Field
Rod rotating about one end: EMF = ½BωL² (L = length of rod)
Rotating disc (radius R): EMF = ½BωR² (disc = collection of infinite rods)
Rotating coil (N turns, area A, angular freq ω):
Φ = NBA cos(ωt); EMF = NBAω sin(ωt) = E₀ sin(ωt)
E₀ = NBAω (maximum EMF = peak EMF)
Definition
- Φ = LI (self inductance L relates flux to current in same coil)
- EMF = −L × dI/dt
- L depends only on geometry (number of turns, area, length) — NOT on current
- Scalar; SI unit = Henry (H); Dimension [ML²T⁻²A⁻²]
- Iron core increases self inductance (μr > 1)
Self inductance of solenoid: L = μ₀n²Al = μ₀N²A/l
With iron core: L = μ₀μrn²Al
Energy stored in inductor: U = ½LI²
Magnetic energy density: u = B²/(2μ₀)
Series inductors: Leq = L₁ + L₂ (no mutual inductance)
Parallel inductors: 1/Leq = 1/L₁ + 1/L₂
Growth and Decay of Current in Inductor (L-R Circuit)
- Growth: I = Imax(1 − e−t/τ) where Imax = ε/R and τ = L/R (time constant)
- Decay: I = I₀e−t/τ
- At t = 0: inductor acts as open wire (current = 0)
- At t = ∞ (steady state): inductor acts as simple wire (maximum current)
- At t = τ: current = 63% of maximum
4
Mutual Inductance and Eddy Currents
Φ₁₂ = MI₂ (flux in coil 1 due to current I₂ in coil 2); similarly Φ₂₁ = MI₁
EMF₁ = −M × dI₂/dt; M depends only on geometry, NOT on current or flux
Reciprocity theorem: M₁₂ = M₂₁ (always true)
For two coils: M = K√(L₁L₂) where K = coupling factor (0 ≤ K ≤ 1)
Series (same flux direction): Leq = L₁ + L₂ + 2M
Series (opposite flux direction): Leq = L₁ + L₂ − 2M
Eddy Currents
- Induced currents in bulk conductor due to changing magnetic flux (like induced current in a loop)
- Cause energy loss as heat (eddy current loss)
- Minimized by using laminated cores (thin insulated sheets)
- Applications: Magnetic braking in trains, Electromagnetic damping, Induction furnace, Electric power meters, Speedometer
5
Alternating Current — Basics
AC vs DC
- DC = Unidirectional (steady or variable); constant direction of current
- AC = Bidirectional; changes direction periodically; sinusoidal form: I = I₀ sin(ωt + φ)
- AC advantages: can be transmitted over long distances; voltage easily stepped up/down by transformer
- AC disadvantages: cannot be stored; skin effect; corona discharge
| Waveform | <I> Full cycle | <I> Half cycle | Irms |
|---|
| Sinusoidal: I₀ sin(ωt) | 0 | 2I₀/π | I₀/√2 |
| Square wave (±I₀) | 0 | I₀ | I₀ |
| Triangular wave (±I₀) | 0 | I₀/2 | I₀/√3 |
| DC = a, AC = b sin(ωt) | a | — | √(a² + b²/2) |
| Full wave rectified | 2I₀/π | 2I₀/π | I₀/√2 |
| Half wave rectified | I₀/π | I₀/2 | I₀/2 |
Key Values for Sinusoidal AC
- <sin ωt>full cycle = 0; <sin ωt>half cycle = 2/π
- <sin²ωt> = <cos²ωt> = ½ (both full and half cycle)
- Irms = I₀/√2 ≈ 0.707 I₀; Vrms = V₀/√2
- Reading of AC ammeter/voltmeter = RMS value (hot-wire type reads both AC and DC)
- Moving coil galvanometer reads only DC
6
AC Circuits — Pure R, L, C
| Circuit | Phase relation | Reactance | Power <P> | At f=0 (DC) | At f=∞ |
|---|
| Pure Resistor | V and I in phase (φ = 0) | R (constant) | IrmsVrms | Acts as R | Acts as R |
| Pure Inductor | V leads I by 90° (φ = +90°) | XL = ωL = 2πfL | 0 (wattless) | XL=0, simple wire | XL=∞, open wire |
| Pure Capacitor | I leads V by 90° (φ = −90°) | XC = 1/(ωC) = 1/(2πfC) | 0 (wattless) | XC=∞, open wire | XC=0, simple wire |
7
Series LCR Circuit and Resonance
Impedance: Z = √(R² + (XL − XC)²)
Phase angle: tan φ = (XL − XC)/R
Power factor: cos φ = R/Z
Average power: <P> = IrmsVrms cos φ = I²rmsR
Resonance: XL = XC → ω₀ = 1/√(LC) → f₀ = 1/(2π√LC)
At resonance: Z = R (minimum); I = maximum; cos φ = 1; <P> = maximum
Three Cases in LCR Series Circuit
- XL > XC: Inductive dominance; voltage leads current; φ > 0
- XL = XC: Resonance; Z = R (minimum); I = maximum; cos φ = 1
- XL < XC: Capacitive dominance; current leads voltage; φ < 0
At resonance: VL = VC (may individually be greater than source V!)
Quality Factor and Bandwidth
- Q factor = ω₀/Δω = ω₀L/R = 1/(ω₀CR) = (1/R)√(L/C)
- Higher Q → sharper resonance peak → more selective circuit
- Bandwidth Δω = ω₂ − ω₁ = R/L
- Half-power frequencies: ω₁, ω₂ = ω₀ ± R/(2L)
⚡ Choke Coil
- Used to control current in AC without significant power loss
- High inductance L, low resistance R → high XL, low R
- Power loss ≈ 0 (cos φ ≈ 0 → wattless current)
- Practical choke: R = low, XL = high, φ ≈ 90° (φ < 90° practically)
Energy Exchange in LC Circuit
- At t = 0: Capacitor fully charged (UE = Q₀²/2C, UB = 0)
- At t = T/4: Capacitor fully discharged; Inductor fully charged (UE = 0, UB = ½LI₀²)
- At t = T/2: Capacitor fully charged again (opposite polarity)
- Energy oscillates between E-field (capacitor) and B-field (inductor)
- Total energy conserved (for ideal LC): Utotal = Q₀²/2C = ½LI₀²
Angular frequency: ω = 1/√(LC)
Time period: T = 2π√(LC)
Analogy: L ↔ mass m; C ↔ 1/k (spring); q ↔ x; I ↔ v
Principle and Working
- Works on principle of Mutual Inductance
- Works only for AC (not DC)
- Used as voltage regulator (step-up or step-down)
- For ideal transformer: Pin = Pout (100% efficiency)
Vs/Vp = Ns/Np = Ip/Is
Power: Pin = VpIp = VsIs = Pout (ideal)
Efficiency: η = Pout/Pin = 1 (100%) for ideal; <1 for practical
| Type | Condition | Voltage | Current |
|---|
| Step-Up transformer | Ns > Np | Vout > Vin | Iout < Iin |
| Step-Down transformer | Ns < Np | Vout < Vin | Iout > Iin |
Power Losses in Transformer
- Copper loss (I²R loss): heat in windings; minimized by low resistance wire
- Eddy current loss: heat in core; minimized by laminated core
- Hysteresis loss: energy in magnetizing/demagnetizing core; minimized by soft iron
- Flux leakage: not all flux links both coils; minimized by design
- Power loss does NOT depend on number of turns
Q: A rod of length 1 m moves with v = 5 m/s ⊥ to B = 0.2 T. Resistance of circuit = 2 Ω. Find induced EMF, current and force on rod.
EMF = BLv = 0.2×1×5 = 1 V; I = EMF/R = 1/2 = 0.5 A; F = BIL = 0.2×0.5×1 = 0.1 N (opposing motion)
Ans: EMF = 1 V; I = 0.5 A; F = 0.1 N ✓
Q: Series LCR circuit: R = 10 Ω, L = 0.1 H, C = 100 μF. Find resonance frequency and Q factor.
f₀ = 1/(2π√LC) = 1/(2π√(0.1×10⁻⁴)) = 1/(2π×10⁻²·⁵) ≈ 50.3 Hz; Q = ω₀L/R = (2π×50.3×0.1)/10 ≈ 0.316
Ans: f₀ ≈ 50.3 Hz; Q ≈ 0.316 ✓
Q: Transformer: 220 V input, 11 V output, input current 0.5 A. Find turns ratio and output current (ideal).
Np/Ns = Vp/Vs = 220/11 = 20; Is = IpVp/Vs = 0.5×220/11 = 10 A
Ans: Turns ratio = 20:1; Iout = 10 A ✓