Electromagnetic Induction & Faraday's Law
Electromagnetic Induction & Faraday's Law
Faraday's discovery that changing magnetic flux induces an electromotive force unified electricity and magnetism. Lenz's law determines the direction of the induced EMF to oppose the change — a statement of energy conservation.
Definition
Faraday's law: \(\mathcal{E} = -d\Phi_B/dt\) where \(\Phi_B = \int\mathbf{B}\cdot d\mathbf{A}\) is the magnetic flux. The negative sign encodes Lenz's law.
Key Result
Mutual inductance: \(\mathcal{E}_2 = -M\,dI_1/dt\). Self-inductance: \(\mathcal{E} = -L\,dI/dt\) with energy stored \(U = \frac{1}{2}LI^2\).
Example 1
A circular loop of radius \(r\) in a uniform \(B(t) = B_0\sin(\omega t)\) has induced EMF \(\mathcal{E} = -\pi r^2 B_0\omega\cos(\omega t)\).
Example 2
A rectangular loop of area \(A\) rotating with angular velocity \(\omega\) in field \(B_0\) generates \(\mathcal{E} = N A B_0\omega\sin(\omega t)\) — the AC generator.
Loading faraday-induction...
Practice
- A coil of 200 turns and area 0.01 m² is in a field changing at 10 T/s. Find the induced EMF.
- Explain the skin effect in a conductor carrying AC current.
- Derive the energy stored in an inductor from Faraday's law.
- Why is eddy current braking dissipative?
Show Answer Key
1. Faraday's law: $\text{EMF} = -N\frac{d\Phi_B}{dt} = -N A\frac{dB}{dt} = -200\times0.01\times10 = -20$ V. Magnitude: 20 V.
2. AC current creates a time-varying magnetic field inside the conductor, inducing eddy currents that oppose the change (Lenz's law). These eddy currents confine the current to a thin surface layer of depth $\delta = \sqrt{2/(\omega\mu\sigma)}$ (skin depth). At higher frequencies, $\delta$ decreases, increasing effective resistance.
3. EMF $= -L\frac{dI}{dt}$. Power input: $P = |\text{EMF}|\cdot I = LI\frac{dI}{dt}$. Energy stored: $U = \int_0^I LI'\,dI' = \frac{1}{2}LI^2$. This energy is stored in the magnetic field: $U = \int \frac{B^2}{2\mu_0}dV$.
4. A moving conductor in a magnetic field induces EMF (motional EMF: $\text{EMF} = Blv$), driving eddy currents. These currents interact with $\mathbf{B}$ via $\mathbf{F}=I\mathbf{l}\times\mathbf{B}$, opposing the motion (Lenz's law). The kinetic energy is converted to Joule heating ($P = I^2R$) in the conductor — irreversible, hence dissipative.