• Ampère’s Law applies in magnetostatics

$$ \oint_C\vec B\cdot \text d \vec l =\mu_0 I_\text{enc} $$

In a non-static situation this equation doesn’t work

Closed surface cuts between plates

Consider the surface $S2$ which cuts between the plates of the capacitor

🕯️ Applying Ampère’s Law

$$ \oint_C\vec B\cdot \text d\vec l=\mu_0\int_{S2}\vec j\cdot\text d\vec A=0 $$

⚠️ This appears to predict $\vec B=0$ even though the surface $S2$ is bound by the same loop $C$

• Thus Ampère’s Law must be incomplete

Displacement current

Maxwell’s insight

• Treat the changing $E$-field like a current

• Using the same example as above, and remembering

$$ E=\frac{Q}{\epsilon_0A}\Rightarrow Q=\epsilon_0AE \quad ; \quad I=\frac{\text d Q}{\text d t}=\epsilon_0A\frac{\partial E}{\partial t} $$

• Thus we get the amended Ampère-Maxwell Law

$$ \oint_C \vec B \cdot \text d \vec l=\mu \left ( I_\text{enc} + \epsilon_0 A \frac {\partial E} {\partial t} \right ) $$

🗒️ This is the special case for plate capacitors

General case for displacement current

$$ \oint_C \vec B \cdot \text d \vec l=\mu \int_S \left ( \vec J + \epsilon_0 \frac {\partial \vec E} {\partial t} \right ) \cdot \text d \vec A $$

🦑 Key point: a changing $E$-field gives rise to a magnetic field