• Faraday’s Law

$$ \mathcal E=-\frac{\text d \Phi_m}{\text dt}=-\frac{\text d}{\text dt}\int_S\vec B\cdot \text d\vec A $$

• True no matter how we change the flux
  • Could change $B$-field
  • Could change size of surface
• A changing magnetic flux always produces EMF but not always current
  • Conducting loop → induced current

Lenz’s Law

The EMF opposes the change

• In the example above flux is decreasing (the current induced creates additional magnetic field so that flux change reduced)

Homopolar generator

Consider a conducting disk of radius $a$ oriented $\perp$ to a field $B$ rotating at $\omega$

$$ \begin{aligned} \text{velocity}\quad \vec v&=r\omega \, \hat \theta \\ \text{Force per unit charge}&=\vec v\times\vec B = vB\, \hat r \\ \text d \mathcal E&=(\vec v \times \vec B)\cdot \text d\vec l \\ &=r\omega B\, \text{d} r \\ \text{Hence} \quad \mathcal E_\text{tot}&=\int^a_0r\omega B \, \text dr \\ &=\frac{1}{2}\omega B a^2 \end{aligned} $$

4$^\text{th}$ Maxwell Equation

• Combining

$$ \mathcal E=-\frac{\text d\Phi_m}{\text d t} \quad ;\quad \Phi_m=\int_S\vec B\cdot\text d\vec A \quad ; \quad \mathcal E=-\oint_C\vec E\cdot\text d\vec l $$

• We find Maxwell’s $4^\text{th}$ equation

$$ \oint_C\vec E\cdot \text d \vec l=- \frac{\text d}{\text dt}\int_S \vec B\cdot \text d\vec A $$

💡 a changing $B$-field creates an $E$-field

Eddy Currents

Consider a metal plate pulled through a $B$-field

• Eddy currents are current loops in extended conducting surfaces
  • Current flow to oppose change in flux
• Exerts forces, use in magnet breaking
• $E_k$ converted to Ohmic heat

EM radiation

• Ampère-Maxwell Law and Faraday’s Law

$$ \begin{aligned} \oint_C\vec B\cdot \text d \vec l &=\mu_0\int_S\left ( \vec j+\epsilon_0\frac{\partial \vec E}{\partial t} \right )\cdot \text d \vec A \\ \oint \vec E\cdot\text d\vec l&=-\frac{\text d}{\text dt}\int_S\vec B\cdot\text d\vec A

\end{aligned} $$

• A change in $E$-field produces a $B$-field
• A change in $B$-field produces a $E$-field