Maxwell Equation

Maxwell’s electromagnetic equations play very important role in deriving electromagnetic wave equations and related properties.

There are four electromagnetic equations. They are:

1. \(\nabla . \vec {D} = \rho\)
2. \(\nabla . \vec B = 0\)
3. \(\nabla \times \vec E = -\frac {\partial \vec {B}}{\partial t}\)
4. \(\nabla \times \vec H = \vec j + \frac {\partial \vec {B}}{\partial t}\)

1. \(\nabla . \vec {D} = \rho\) (gauss law for electric field):

   Assume a surface S bounding a volume V in a dielectric medium. According to Gauss law, the integral \(\int \vec E. d\vec S \) of normal component of \(\vec E \) over any closed surface is equal to total charge enclose within surface.
   Total charge =Free charge + Polarised charge
   \begin{align*} \rho_{total} = \rho + \rho ‘ \end{align*}
   $$\rho= \text {free charge density at a point in small volume element dv} $$
   $$\rho’ = \text {Polarised charge density} = -\nabla P $$
   \begin{align*} \rho_{total} &= \rho - \nabla P \\ \int _s E.ds &= q =\int \frac {1}{\epsilon_o}(\rho_{total} )\\ &= \int \frac {1}{\epsilon_o}(\rho - \nabla P) \\ \end{align*}
   Gauss divergence theorem
   \begin{align*} \int_s \vec E.d\vec s &= \int_v (\nabla . \vec E)dv \\ \int_v (\nabla . \vec E)dv &= \int \frac {1}{\epsilon_o}(\rho - \nabla P)dv \\ \text {or,}\: \int _v \nabla (\epsilon_o\vec E+ P)dv &= \int \rho dv \end{align*}
   \begin{align*}\text {the quantity electric displacement} \vec D = \epsilon _o \vec E + \vec P\\\epsilon_0 &= \text {dielectric constant}\\ P &= \text {polarization vector}\\\int (\nabla . \vec D)\: dv &= \int \rho dv \\ \int (\nabla . \vec D - \rho)dv = 0 \\ dv\neq 0 \:\: \text {So},\: \nabla . D -\rho = 0\\\text {i.e} \nabla . D =\rho \end{align*}
   In isotropic medium, \(\vec D,\: \vec E\:\text {and}\: \vec P\) have same direction, for small field \(\vec D\) \propto \vec E\: \text {i.e.}\: \vec B = \epsilon_o\vec E\)
   $$\text {i.e}\: \nabla \vec E = \frac {1}{\epsilon_o \rho$$
   
   

2. \(\nabla . B = 0\) (Gauss law for magnetic field

   magnetic lines are either closed or goes upto \(\infty \). Number of magnetic lines entering in closed surface is equal to number of magnetic lines of force leaving. Magnetic flux B in closed surface is zero i.e. \(\int B. d\vec s = 0\)
   Gauss divergence theorem
   \begin{align*} \int_s \vec B.d\vec s &= \int_v (\nabla . \vec B)dv \end{align*}
   since, the arbitrary volume is considered, so integral should vanish for boundary surface
   $$ \nabla . \vec B = 0$$

3. \(\nabla \times \vec E = -\frac {\partial \vec B}{\partial t}\) (Faraday Law of Electromagnetic Equation)

   According to the Faraday law of electromagnetic induction, the induced emf is given as
   \begin{align*} E_{emf} = -\frac {\partial \phi}{\partial t} = -\frac {\partial }{dt}\int \vec B.d\vec s\dots (1) \end{align*}
   \( \phi = \int \vec B. d\vec s \) is magnetic flux in a closed loop of area S. sine emf is work done by unit charge. So, emf in closed loop as
   \begin{align*} E_{emf} &= \oint \vec E. \partial \vec l = \int (\nabla \times E)d\vec s \dots (2) \\ \text {From equation}\:(1)\: \text {and}\:(2) \\ \int (\nabla \times E)d\vec s &= -\frac {\partial }{dt}\int \vec B.d\vec s \\ \int \left (\nabla \times E + \frac {\partial \vec B}{dt} \right )d\vec s &= 0\\ \text {Since}\: ds \neq 0 \\ \nabla \times E + \frac {\partial \vec B}{dt} = 0 \\ \nabla \times E &=- \frac {\partial \vec B}{dt} \end{align*}

4. \(\nabla \times \vec H = \vec j + \frac {\partial \vec {B}}{\partial t}\) (Generalized Ampere’s law)

   According to Ampere’s circuital law, the work done in carrying a unit magnetic pole in a closed loop associated with unit current.
   \begin{align*}\oint \vec H .d\vec lL = I \int_s \vec J .ds \dots (1) \\\text {Stroke}'s \:\text {theorem}\\ \int (\nabla \times \vec H.ds &=\oint H.dl \\ \text {For arbitrary volume}\\ \nabla \times \vec H &= \vec J \dots (8) \end{align*}
   This relation is derived on the basis of Ampere’s law valid for steady current. In case of changing electric field.
   \begin{align*} \nabla .(\nabla \times \vec H) &= \nabla.\vec J \\ \nabla \vec J &= 0 \\ \text {which contradicts to}\: \nabla . \vec J - \frac {\partial \rho}{\partial t}\\ \text {so} \nabla \times \vec H &= \vec J + \vec J' \\ \text {Again divergence} \\ \nabla .(\nabla \times \vec H) &= \nabla (\vec J + \vec J' )\\ \text {or,}\: \nabla .\vec J + \nabla. \vec J &= 0 \\ \text {or,}\: \nabla .\vec J &= -\nabla . \vec J = -\frac {\partial \rho}{\partial t}\end{align*}
   From Maxwell’s first equation that \(\rho = \nabla . \vec D\), the above relation becomes
   \begin{align*} \nabla .\vec J &= \frac {\partial}{\partial t}(\nabla . \vec D) = \nabla. \frac {\partial \vec D}{\partial t} \\ \vec J &= \nabla. \frac {\partial \vec D}{\partial t} \end{align*}
   Equation 1 can be modified as
   $$\nabla \times H = \vec J +\frac {\partial \vec D}{\partial t} $$

Bibliography

P.B. Adhikari, Bhoj Raj Gautam, Lekha Nath Adhikari. A Textbook of Physics. kathmandu: Sukunda Pustak Bhawan, 2011.

Jha, V. K.; 'Lecture title'; Maxwell's Equation; St. Xavier's College, Kathmandu; 2016.