AP E&M Chapter 14-16 Part 4: Maxwell’s Equations
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💡 Ampere’s Law is not Perfect
We take the divergence of both sides of the formula of Ampere’s Law:
\[\begin{aligned} \nabla \times \vec{B} &= \mu_0 \vec{J} \\ \nabla \cdot (\nabla \times \vec{B}) &= \mu_0 (\nabla \cdot \vec{J}) \\ \nabla \cdot \vec{J} &= 0 \end{aligned}\]This result is inconsistent with the law of charge conservation (电荷守恒定律):
\[\boxed{ \frac {\partial \rho} {\partial t} + \nabla \cdot \vec{J} = 0 }\]Therefore, Maxwells added a term into the formula:
\[\colorbox{#424242}{$ \color{white}\boxed{ \nabla \times \vec{B} = \mu_0 (\vec{J} + \varepsilon_0 \frac {\partial \vec{E}} {\partial t}) } $}\]We can verify the validity by taking the divergence on both sides again:
\[\begin{aligned} \nabla \times \vec{B} &= \mu_0 (\vec{J} + \varepsilon_0 \frac {\partial \vec{E}} {\partial t}) \\ \nabla \cdot (\nabla \times \vec{B}) &= \mu_0 (\nabla \cdot \vec{J} + \varepsilon_0 \frac {\partial (\nabla \cdot \vec{E})} {\partial t}) \\ \nabla \cdot \vec{J} + \frac {\partial \rho} {\partial t} &= 0 & \nabla \cdot \vec{E} = \frac \rho {\varepsilon_0} \\ \end{aligned}\]Therefore, the new Ampere’s Law is consistent with charge conservation.
Notably, the added term is not unique, as long as it does not affect the divergence. For example, $\varepsilon_0 \frac {\partial \vec{E}} {\partial t} + \nabla \times \vec{F}$ (where $\vec{F}$ is any vector field) is also valid. However, the correctness of this particular formula is guaranteed by experiment (which means this is a law), and therefore this formula is added into the Maxwell’s Equations.
The name of this new law is Ampere-Maxwell Law.
Ampere-Maxwell Law
The integral form of Ampere-Maxwell Law is:
\[\colorbox{#424242}{$ \color{white}\boxed{ \displaystyle \oint \vec{B} \cdot d\vec{l} = \mu_0 (I_{\text{enc}} + \varepsilon_0 \frac {\partial \Phi_E} {\partial t}) } $}\]The term $\varepsilon_0 \frac {\partial \Phi_E} {\partial t}$ is called displacement current (位移电流).
Maxwell’s Equations
Maxwell’s Equations are the four fundamental equations of electricity and magnetism:
- Gauss’s Law for Electricity
- Gauss’s Law for Magnetism
- Faraday’s Law
- Ampere-Maxwell Law
💡This is the differential form of these four equations:
\[\colorbox{#424242}{$ \color{white}\boxed{ \begin{cases} \nabla \cdot \vec{E} = \frac \rho {\varepsilon_0} \\ \nabla \cdot \vec{B} = 0 \\ \nabla \times \vec{E} = - \frac {\partial \vec{B}} {\partial t} \\ \nabla \times \vec{B} = \mu_0 (\vec{J} + \varepsilon_0 \frac {\partial \vec{E}} {\partial t}) \\ \end{cases} } $}\]Combined with the Lorentz force formula, these 5 equations can explain every problem in electricity and magnetism.
\[\colorbox{#424242}{$ \color{white}\boxed{ \vec{F} = q (\vec{E} + \vec{v} \times \vec{B}) } $}\]With the discovery of Maxwell’s equations, classical electromagnetism reached its end.