Electromagnetic Induction – Study Notes (MIT‑Level)
Audience – Advanced undergraduate / beginning graduate physics students
Goal – Derive Faraday’s law and the related concepts from first principles, explore non‑trivial edge cases, and provide a challenging problem that tests a deep understanding of the theory.
1. Introduction
Electromagnetic induction underlies almost every modern electrical technology: transformers, motors, generators, inductors, and magnetic storage. The phenomenon is a direct consequence of the time‑dependence of the magnetic field and the geometrical/topological properties of the circuit.
Key Insight
The electromotive force (EMF) induced in a closed loop is equal to the negative time‑rate of change of magnetic flux through that loop. This statement is a manifestation of Faraday’s law, which is itself a consequence of the Maxwell–Faraday equation.
2. Maxwell’s Equations in Differential Form
We work in SI units, with c = 1/√(με) for convenience (but keep μ₀ and ε₀ explicit for clarity):
| Equation | Symbol | Physical Meaning |
|---|---|---|
| Gauss’s Law for E | ∇·E = ρ/ε₀ | Electric field divergence equals charge density |
| Gauss’s Law for B | ∇·B = 0 | No magnetic monopoles |
| Faraday–Maxwell Law | ∇×E = –∂B/∂t | Time‑varying B produces curl in E |
| Ampère–Maxwell Law | ∇×H = J + ∂D/∂t | Currents and changing electric fields produce curl in H |
Differential vs. Integral forms:
Integral equations are obtained by applying the divergence and Stokes’ theorems to the differential equations. The integral form of Faraday’s law is the starting point for induction.
3. Faraday’s Law (Integral & Differential)
3.1 Integral Form
For a surface (S) bounded by a closed curve (C):
[ \boxed{\displaystyle \oint_{C}\mathbf{E}\cdot d\mathbf{l} ;=; -\frac{d}{dt}\int_{S}\mathbf{B}\cdot d\mathbf{A}} ]
- Left side: EMF around the loop.
- Right side: Negative time‑rate of change of magnetic flux (\Phi_B).
3.2 Differential Form
Apply Stokes’ theorem to the left side:
[ \oint_{C}\mathbf{E}\cdot d\mathbf{l} = \int_S (\nabla\times\mathbf{E})\cdot d\mathbf{A} ]
Set equal to the RHS:
[ \nabla\times\mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} ]
This is the Maxwell–Faraday equation.
4. Motional EMF
When a conductor moves through a static magnetic field, charges experience the Lorentz force:
[ \mathbf{F} = q(\mathbf{v}\times\mathbf{B}) ]
The work done per unit charge along a path (dl) gives an induced electric field:
[ \mathbf{E}_{\text{mot}} = \mathbf{v}\times\mathbf{B} ]
Integrating around a closed loop yields:
[ \mathcal{E} = \oint \mathbf{E}_{\text{mot}}\cdot d\mathbf{l} = \oint (\mathbf{v}\times\mathbf{B})\cdot d\mathbf{l} = -\frac{d\Phi_B}{dt} ]
Why the minus sign?
The EMF acts to oppose the change in flux (Lenz’s law), ensuring conservation of energy.
5. Lenz’s Law: Direction of Induced Current
Lenz’s law states that the induced current will flow such that its own magnetic field opposes the change in the external flux. Mathematically:
[ \mathcal{E} = -\frac{d\Phi_B}{dt} ]
- Increase in (\Phi_B) → induced current produces a magnetic field opposing the increase.
- Decrease in (\Phi_B) → induced current produces a magnetic field aiding the decrease.
6. Self‑Inductance & Mutual Inductance
6.1 Self‑Inductance
For a single circuit with flux (\Phi = L I):
[ L = \frac{\Phi}{I} ]
Induced EMF:
[ \mathcal{E} = -L \frac{dI}{dt} ]
Energy stored in the magnetic field:
[ U = \frac{1}{2} L I^2 = \frac{1}{2}\int \frac{B^2}{\mu_0} dV ]