Maxwell

The derivation of Maxwell’s equations in the differential (or point) form is based on applying fundamental vector calculus theorems (Gauss’s Divergence Theorem and Stokes’ Theorem) to the integral forms of the four primary laws of electromagnetism.

I. Maxwell’s First Equation (Gauss’s Law for Electrostatics)

This equation describes the origin of the electric field from charge density.

1. Start with the Integral Form of Gauss’s Law: Gauss’s law for electrostatics states that the total electric flux ($\psi$) crossing a closed surface ($S$) is equal to the total charge ($q$) enclosed by that surface. In a dielectric medium, the charge is the sum of free charge density ($\rho$) and polarized charge density (${\rho }_p$).

2. ${\oint }_S{ϵ}_0\vec{E}\cdot d\vec{s}={\int }_V\frac{1}{{ϵ}_0}(\rho +{\rho }_p)dV$ The source defines $q={\int }_V\rho dV$ and notes that ${\rho }_p=-\nabla \cdot \vec{P}$.

3. Apply Divergence Theorem: The Divergence Theorem is used to convert the surface integral on the left side into a volume integral:

4. ${\oint }_S{ϵ}_0\vec{E}\cdot d\vec{s}={\int }_V\nabla \cdot ({ϵ}_0\vec{E})dV$ Equating the volume integrals:

5. ${\int }_V\nabla \cdot ({ϵ}_0\vec{E})dV={\int }_V(\rho -\nabla \cdot \vec{P})dV$ This can be rearranged as:

6. ${\int }_V\nabla \cdot ({ϵ}_0\vec{E}+\vec{P})dV={\int }_V\rho dV$

7. Introduce Electric Displacement ($\vec{D}$): The Electric Displacement vector is defined as $\vec{D}={ϵ}_0\vec{E}+\vec{P}$.

8. ${\int }_V\nabla \cdot \vec{D}dV={\int }_V\rho dV$

9. Differential Form: Since the volume $V$ is arbitrary, the integrands must be equal:

10. $\nabla \cdot \vec{\mathbf{D}}=\rho$

II. Maxwell’s Second Equation (Gauss’s Law for Magnetostatics)

This equation states that there are no magnetic monopoles.

1. Start with the Integral Form: Gauss’s Law for Magnetostatics states that the total magnetic flux passing through any arbitrary closed surface ($S$) is zero. This is because magnetic field lines are closed (or go off to infinity), meaning the flux entering the surface equals the flux leaving.
3. ${\oint }_S\vec{B}\cdot d\vec{s}=0$
4. Apply Divergence Theorem: Using the Gauss Divergence theorem to convert the surface integral to a volume integral:
6. ${\int }_V\nabla \cdot \vec{B}dV=0$
7. Differential Form: Since the volume $V$ is arbitrary:
9. $\nabla \cdot \vec{\mathbf{B}}=0$

III. Maxwell’s Third Equation (Faraday’s Law of Induction)

This equation shows that a time-varying magnetic field produces an electric field.

1. Start with Faraday’s Law: The law states that the induced electromotive force ($e$) in a closed loop is the negative rate of change of the magnetic flux ($\Phi$):
3. $e=-\frac{d\Phi }{dt}$ The electromotive force is the line integral of the electric field around the closed loop $C$ ($e={\oint }_C\vec{E}\cdot d\vec{l}$), and the magnetic flux is the surface integral of the magnetic field ($\Phi ={\iint }_S\vec{B}\cdot d\vec{s}$). Equating these gives the integral form:
5. ${\oint }_C\vec{E}\cdot d\vec{l}=-\frac{d}{dt}{\iint }_S\vec{B}\cdot d\vec{s}$
6. Apply Stokes’ Theorem: Stokes’ theorem is used to convert the contour integral ($\oint \vec{E}\cdot d\vec{l}$) into a surface integral of the curl of $\vec{E}$:
8. ${\iint }_S(\nabla \times \vec{E})\cdot d\vec{s}=-{\iint }_S\frac{\partial \vec{B}}{\partial t}\cdot d\vec{s}$ (For a stationary surface, the time derivative $d/dt$ can be moved inside the surface integral and becomes a partial derivative $\partial /\partial t$).
9. Differential Form: Since the equality must hold for any surface $S$:
11. $\nabla \times \vec{\mathbf{E}}=-\frac{\partial \vec{\mathbf{B}}}{\partial t}$

IV. Maxwell’s Fourth Equation (Ampere-Maxwell Law)

This equation shows that both electric currents and changing electric fields produce magnetic fields.

1. Start with Ampere’s Circuital Law (Steady Current): For steady currents, the integral form is ${\oint }_C\vec{H}\cdot d\vec{l}=I$. Using Stokes’ theorem and the definition of current density ($\vec{J}$), this leads to the differential form:
3. $\nabla \times \vec{H}=\vec{J}$
4. Necessity of Displacement Current: This equation is valid only for steady currents. Taking the divergence of this equation, $\text{div}(\nabla \times \vec{H})=\text{div}\vec{J}$. Since the divergence of a curl is always zero, this implies $\text{div}\vec{J}=0$, which contradicts the continuity equation for time-varying fields: $\text{div}\vec{J}=-\frac{\partial \rho }{\partial t}$.
5. Maxwell’s Correction: Maxwell suggested correcting this law by introducing an additional current density term, ${\vec{J}}_D$, such that the total current density ${\vec{J}}_T$ satisfies the continuity equation ($\text{div }{\vec{J}}_T=0$).
7. $\text{div}(\vec{J}+{\vec{J}}_D)=0⟹\text{div }{\vec{J}}_D=-\text{div }\vec{J}$
8. Deriving ${\vec{J}}_D$:
   • Substitute $\text{div }\vec{J}=-\frac{\partial \rho }{\partial t}$ into the above relation:
   • $\text{div }{\vec{J}}_D=\frac{\partial \rho }{\partial t}$
   • From Maxwell’s First Equation, we use $\rho =\nabla \cdot \vec{D}$.
   • $\text{div }{\vec{J}}_D=\frac{\partial }{\partial t}(\nabla \cdot \vec{D})$
   • Assuming spatial and temporal differentiation commutes:
   • $\text{div }{\vec{J}}_D=\nabla \cdot \left(\frac{\partial \vec{D}}{\partial t}\right)$
   • This shows the additional term, known as the displacement current density, must be:
   • ${\vec{J}}_D=\frac{\partial \vec{D}}{\partial t}$
9. Differential Form (Point Form): Substituting the total current density ${\vec{J}}_T=\vec{J}+{\vec{J}}_D$ back into
11. the curl equation: $\nabla \times \vec{\mathbf{H}}=\vec{\mathbf{J}}+\frac{\partial \vec{\mathbf{D}}}{\partial t}$