📂Electrodynamics

Maxwell's Equations

Formulas

Maxwell’s Equations

$(\text{i}) \quad \nabla \cdot \mathbf{E}=\dfrac{1}{\epsilon_{0}}\rho$ (Gauss’s Law)

$(\text{ii}) \quad \nabla \cdot \mathbf{B}=0$ (Gauss’s Law for Magnetism)

$(\text{iii}) \quad \nabla \times \mathbf{E} = -\dfrac{\partial \mathbf{B}}{\partial t}$ (Faraday’s Law)

$(\text{iv}) \quad \nabla \times \mathbf{B} = \mu_{0} \mathbf{J}+\mu_{0}\epsilon_{0}\dfrac{\partial \mathbf{E}}{\partial t}$ (Ampère’s Law)

Description¹

Before Maxwell completed the Maxwell’s equations, the four equations concerning the electric field and the magnetic field were as follows:

$(\text{i}) \quad \nabla \cdot \mathbf{E}=\dfrac{1}{\epsilon_{0}}\rho$

$(\text{ii}) \quad \nabla \cdot \mathbf{B}=0$

$(\text{iii}) \quad \nabla \times \mathbf{E} = -\dfrac{\partial \mathbf{B}}{\partial t}$

$(\text{iv}) \quad \nabla \times \mathbf{B} = \mu_{0} \mathbf{J}$

Theoretically, nearly all of electromagnetism can be explained with just these four equations concerning the divergence and the curl of the electric and magnetic fields. They weren’t grouped together without reason. However, there was a significant error in the above formula $\text{(iv)}$. Since the divergence of curl is always $0$, taking the divergence of $(\text{iv})$ results in the following.

$$ \begin{equation} 0 = \nabla \cdot (\nabla \times \mathbf{B})=\mu_{0} (\nabla \cdot \mathbf{J}) \ne 0 \end{equation} $$

Here is where a problem arises. While Ampère’s Law holds well with a constant current, making the right side $0$, it is not generally the case.

To make the right side $0$, an idea comes from using the continuity equation and Gauss’s law to change the right side:

$$ \nabla \cdot \mathbf{J}=-\dfrac{\partial \rho}{\partial t}=-\dfrac{ \partial( \epsilon_{0} \nabla \cdot \mathbf{E})}{\partial t}=-\nabla \cdot \left(\epsilon_{0} \dfrac{\partial \mathbf{E}}{\partial t } \right) $$

Therefore, using $\mathbf{J}+\epsilon_{0}\dfrac{\partial \mathbf{E} }{\partial t}$ instead of $\mathbf{J}$ can make the right side of $(1)$ become $0$. The corrected $\text{(iv)}$ is as follows:

$$ \text{(iv)} \quad \nabla \times \mathbf{B} = \mu_{0}\mathbf{J} + \mu_{0}\epsilon_{0}\dfrac{\partial \mathbf{E}}{\partial t} $$

The modified formula still satisfies magnetostatics, correcting the faulty part without violating the existing laws. Indeed, there is a reason this was corrected later by Maxwell. Many laws of electromagnetism were discovered and proven through experiments. However, the difference in magnitudes between the two terms of the above formula is usually so significant that it was incredibly hard to discover experimentally.

$$ \left| \epsilon_{0}\dfrac{\partial \mathbf{E}}{\partial t} \right| \ll \left| \mathbf{J} \right| $$

The formula modified by Maxwell encapsulates the idea that 'a changing electric field generates a magnetic field'. This was confirmed in 1888 by Hertz’s electromagnetic wave experiment.