Integral form of the Maxwell’s Equations
Lets have a look at the integral form of the Maxwell’s Equations
Gauss’ law for Electrostatics
Gauss’ law for Electrostatics states that the total electric flux through a closed surface is proportional to the enclosed electric charge. Mathematically, this can be expressed as
∇. E = ρ / ε0
Where E is the electric field, ds is the infinitesimal area element and the closed integral of E over ds gives the electric flux.
Gauss’ law for Magnetism
This law states that the total magnetic flux through a closed surface is zero. This basically means that magnetic monopoles cannot exist. We know that when a magnet is broken down into smaller pieces, each piece will have its own north and south pole – one cannot isolate a single pole. Magnetic poles always exist in pairs and there is no net magnetic field outflow through a closed surface. Mathematically this can be written as
∇. B = 0
Where B is the magnetic field, ds is the infinitesimal area element and the closed integral of B over ds gives the magnetic flux.
Faraday’s law of Induction
Faraday’s law states that changing magnetic flux always generates an EMF that is equal to the negative of the rate of change of the magnetic flux enclosed by the path. The emf is basically the voltage that can be obtained from integrating the electric field. The mathematical form of the Maxwell-Faraday law becomes
∇ × E = − ∂B/ ∂t
This law of electromagnetic induction is the source of all power – the operating principle of electric generators.
Ampere’s Circuital Law
Ampère’s circuital law relates the magnetic field around a closed loop to the electric current passing through the loop. Although named after Ampère, it was actually derived by Maxwell using hydrodynamics(study of liquid flow) in 1861.
∇ × B = μ0J + μ0ε0 ∂E/ ∂t
Where B is the magnetic field, dl is the infinitesimal line element and the line integral of B gives the current flowing through the wire.
Maxwell added a term to this equation known as the displacement current that was defined by the rate of change in electric field. Its origin is not due to actual charge flow, but due to changing electric field. Defined mathematically as,
ID = ε0 A . ∂E/∂t
Where ID is the displacement current, E is the electric field, and A is the surface area.
A prominent example where this comes to play is the case of capacitors. There is no charge transfer between the plates of a capacitor when it first begins to charge. The modified Ampère-Maxwell circuital law now becomes
∮[Tex]\bold{\overrightarrow{\rm B}}[/Tex] . dl = μ0(I + ID)
Maxwell’s Equations in Electromagnetism
Maxwell’s Equations are a set of four equations proposed by mathematician and physicist James Clerk Maxwell in 1861 to demonstrate that the electric and magnetic fields are co-dependent and two distinct parts of the same phenomenon known as electromagnetism.
These formulas show how variations in the quantity or velocity of charges can impact magnetic and electric fields. Maxwell went on to establish that light is an electromagnetic wave caused by oscillations in the electric and magnetic fields. Maxwell’s equations give a mathematical model for the operation of all electronic and electromagnetic devices, ranging from power generation to wireless communication.
Table of Content
- What are Maxwell’s Equations of Electromagnetism?
- Maxwell’s First Equation
- Maxwell’s Second Equation
- Maxwell’s Third Equation
- Maxwell’s Fourth Equation
- Applications of Maxwell Equations