Magnetic Flux
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Magnetic flux through a small area dA in a magnetic field B:
dϕ = B · dA = B dA cosθ
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Flux depends on angle θ:
- θ = 90° ⇒ ϕ = 0 (surface parallel to magnetic field)
- 0° ≤ θ ≤ 90° ⇒ positive flux
- 90° ≤ θ ≤ 180° ⇒ negative flux
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Magnetic flux through a closed surface is zero:
∮ B · dA = 0(No magnetic monopoles)
Faraday’s Laws of Electromagnetic Induction
- First Law: Change in magnetic flux induces an EMF in a circuit.
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Second Law: Induced EMF is proportional to rate of change of flux:
e ∝ dϕ/dtFor N turns:e = −N dϕ/dt
Lenz’s Law
- The direction of induced current opposes the cause producing it.
- Based on the law of conservation of energy.
Important Results
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Induced EMF is independent of resistance:
e = −dϕ/dt
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Induced current depends on resistance:
I = e / R
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Total induced charge depends only on change in flux:
q = (ϕ₂ − ϕ₁) / R
Induced EMF in Conducting Rod
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Rod moving in uniform magnetic field:
|e| = B l v sinθ
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Rod rotating in magnetic field:
e = (1/2) B ω l² = B A f
Self-Inductance
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Magnetic flux proportional to current:
ϕ = L I
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Induced EMF:
e = −L (dI/dt)
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Self-inductance of solenoid:
L = μ₀ μᵣ N² A / l
Mutual Inductance
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Flux in secondary due to current in primary:
ϕ = M I
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Induced EMF:
e = −M (dI/dt)
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Mutual inductance of coaxial solenoids:
M = μ₀ μᵣ N₁ N₂ A / l
Combination of Inductors
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Series (no mutual induction):
L = L₁ + L₂
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Series (with mutual induction):
L = L₁ + L₂ ± 2M
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Parallel:
1/L = 1/L₁ + 1/L₂
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Coupling coefficient:
M = K √(L₁ L₂)
LR Circuit
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Growth of current:
I = (E/R)(1 − e−Rt/L)
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Decay of current:
I = (E/R) e−Rt/L
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Time constant:
τ = L / R
Transformer
- Works on mutual induction; used only for AC.
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Voltage relation:
Vₛ / Vₚ = Nₛ / Nₚ
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Ideal transformer:
Vₚ Iₚ = Vₛ Iₛ
- Step-up: Nₛ > Nₚ | Step-down: Nₛ < Nₚ
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Last modified: December 14, 2025
