Maxwell’s Equations
- Post by: radha
- February 28, 2022
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This post is about Modified Maxwell’s equations for EM wave propagation. We will see how Electric field and Magnetic field depends on each other .
Maxwell’s Equation in static field
Maxwell’s equations explain, the basic behavior of EM fields at every point (Microscopic form) and also over a region (Macroscopic form).
Maxwell’ Equation in Static Fields
We will see Maxwell’s equations in integral form as well as in point form respectively.
INTEGRAL FORM
1.$\oint\vec{D}\cdot\vec{d s}= Q$
2. $\oint{E}\cdot{d l}=0$
3. $\oint {B}\cdot{d s}= 0$
4. $\oint{H}\cdot{d l}={I}$
POINT FORM
1. $\nabla\cdot{D}= \rho_{v}$
2. $\nabla \times{B}=0$
3. $\nabla \cdot{B}= 0$
4. $\nabla\times {H}={J}$
Equation 2 and 4 are modified, since the concepts of AC voltage and AC currents are included in these equations only.
The surface integrals and divergence expression are consistently the same in Static or Time varying Fields. However the line integrals and curls expression are modified to explain AC voltage and AC currents.
NOTE:
If $\nabla\cdot{B}=0$ ;
then $\nabla\cdot(\mu{H})=0;$ only if $\mu$ = Constant
$\nabla\times{H}=0$ If $\mu= Constant$,( Medium is Isotropic or Medium is Homogeneous)
Similarly,
$\nabla \times{H}=0$
$\nabla\times{D}=0$
This is also correct if $\in= Constant$ (Medium is isotropic/ medium is Homogeneous)
Analysis:
First we Concept of Open Integrals:
1.$\int\vec{D}\cdot\vec{d s}= \psi_{e}{(Coulombs})$
2.$\int{E}\cdot{d l}= \text{Electro motive Force(EMF)}{(Volts)}$
3.$\int {B}\cdot{d s}= \psi_m(webers)$
4.$\int{H}\cdot{d l}=\text{Magnetic motive Force(MMF)}(Ampere)$
These are NOT MAXWELL’s EQUATIONS.
Coulombs/sec gives current (Ampere) and Weber’s/ sec gives Voltage. Hence Electric and Magnetic Fields interact with each other with respect to time.
FARADAY’S LAW AND MAXWELL’S SECOND EQUATION:
STATEMENT:
There is a EMF induced even in a closed conductor when a magnetic flux crossing the surface changes with time.
Rate of change of Magnetic Flux with time is induced EMF.
Mathematically,
$\oint{E}\cdot{d l}=-\frac{d\psi_m}{dt}$
this negative sign is due to LENZ LAW. The induced EMF(effect) always opposes the changing Flux (Cause).
In actual $\oint{E}\cdot{d l}=0$ ; when there is no change in magnetic flux. But when the magnetic flux changes it induces an EMF in the conductor and the conductor opposes this changes . This gives LENZ LAW.
Hence;
$\oint{E}\cdot{d l}=-\frac{d\psi_m}{dt}= -\frac{d\int{B}\cdot{d s}}{dt}$
$\oint{E}\cdot{d l}=\iint-\frac{dB}{dt}\cdot{d s}$
or,
$\iint(\nabla\times{E})\cdot{d s}=\iint-\frac{dB}{dt}\cdot{ds}$
Hence,
$\oint{E}\cdot{dl}=\int-\frac{dB}{dt}\cdot{d s}$
$\nabla\times{E}= -\frac{dB}{dt}$
This is modified Maxwell’s second Equation.
Potential is Unique at a given point at a given time, but can vary with time. Hence the modification is needed.
Inconsistency of Ampere’s Law and Maxwell’s fourth Equation:
Conduction current flows in the wires of the circuit shown. But between the plates of the capacitor, conduction current can not flow in between the plates of capacitor the displacement current Id flow. If Id is assumed to be zero and only if only conduction current (Ic) flows , then the circuit will not be complete . This is Inconsistency of Ampere’s Law.
Mathematically,
${c}=\frac{dQ}{dV}$
${c}=\frac{dQ}{dt}\cdot\frac{dt}{dV}$
so, $I_c={c}\frac{dV}{dt}$
NOTE: A capacitor is a linear element obeying OHM’s LAW when applied with a hormonic voltage . Hence there is a current between the wires and plates but there is no flow between the plates. This inconsistency of closed loop current is explained by Maxwell as:
$\oint{H}\cdot{dl}= {I_c}+{I_d}$
$\nabla\times{H}={J_c}+{J_d}$
using continuity equation for Jd we get
$\nabla\cdot{J_d}= \frac{d\rho_v}{dt}$
$\nabla\cdot{J_d}=\frac{d}{dt}(\nabla\cdot{D})$
$\nabla\cdot{J_d}=\nabla\cdot\frac{d}{dt}{D}$
so,
${J_d}=\frac{dD}{dt}$
${I_d}=\int\frac{dD}{dt}\cdot{ds}$
hence, modified Maxwell’s fourth equation is:
$\oint{H}\cdot{dl}= {I_c}+{I_d}= {I_c}+\int\frac{dD}{dt}\cdot{ds}$
$\nabla\times{H}={J_c}+\frac{dD}{dt}$
When an AC voltage is applied to capacitor plates, they charge and discharge automatically which is in itself a continuous process . However as the $\rho_s$ on the plates change changes with time the electric flux density D changes with time.
This changing Electric Flux is also a form of current that is,
$\frac{dD}{dt}= {J_d} ={Coulombs/m^2.s}=Amp/m^2$
Where Jd displacement current density.
Hence $J_d =\in\frac{dE}{dt}$ is wireless current in free space due to advancement Electric Field.
$J_c={\sigma}{E}$ wired current in conductors due to electron movement
$J_d =\in\frac{dE}{dt}$ wireless current
This wireless current Id(or Jd) is the cause of flow of EM or propagation of EM waves in free space.
SUMMARY:
$\oint{H}\cdot{dl}= {I_c}+\int\frac{dD}{dt}\cdot{ds}$
A time varying Electric Flux is a format of current.
$\oint{E}\cdot{dl}=\int-\frac{dB}{dt}\cdot{d s}$
A time varying magnetic flux is a cause of voltage.
SUMMARY OF POINT FORM:
Time varying E field always produces space varying Magnetic field and vice-versa.
$\nabla\times{E}= -\frac{dB}{dt}$
$\nabla\times{H}={J_c}+\frac{dD}{dt}$
Accumulation leads to flow and flow develops an accumulation. Accumulation and flow depends on the material constants that are \mu and E and \sigma.
$\nabla\times{E}=-{\mu}\frac{dH}{dt}$
$\nabla\times{H}={\sigma}{E}+{\in}\frac{dE}{dt}$
Material constants are:
$\in=$ Farads /m -E Field holding ability.
$\mu=$ Henry /m – H field holding ability.
$\sigma= \mho/m$- Allowing current flow.
The E to H transformation or vice versa always depends on the material constant that is “ THE PERMITTIVITY ABILITIES OF THE MATERIAL” for different energy formats ; is critically important. The movement of E/ H wave, the time variations should be some special signals. The differentiation (derivative) of taken E and H should result in same E and H that is HORMONICS FUNCTION.