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Derive the four maxwell equations of electromagnetism waves in integral form

2 months ago

Maxwell First Equation

Maxwell first equation is based on the Gauss law of electrostatic which states that “when a closed surface integral of electric flux density is always equal to charge enclosed over that surface”

Mathematically Gauss law can be expressed as,

Over a closed surface the product of electric flux density vector and surface integral is equal to the charge enclosed.

∯D⃗ .ds¯=Qenclosed ——(1)

Any closed system will have multiple surfaces but a single volume. Thus, the above surface integral can be converted into a volume integral by taking the divergence of the same vector. Thus, mathematically it is-

∯ D¯.ds¯=∭▽.D⃗ dv⃗ —-(2)

Thus, combining (1) and (2) we get-

Charges in a closed surface will be distributed over its volume. Thus, the volume charge density can be defined as –

ρv=dQdv measured using C/m3

On rearranging we get-

dQ=ρvdv

On integrating the above equation we get-

Q=∭ρvdv ——-(4)

The charge enclosed within a closed surface is given by volume charge density over that volume.

Substituting (4) in (3) we get-

∭▽.Ddv=∭ρvdv

Canceling the volume integral on both the sides, we arrive at Maxwell’s First Equation-

⇒▽.Ddv=ρv

Maxwell Second Equation

Maxwell second equation is based on Gauss law on magnetostatics.

Gauss law on magnetostatics states that “closed surface integral of magnetic flux density is always equal to total scalar magnetic flux enclosed within that surface of any shape or size lying in any medium.”

Mathematically it is expressed as –

∯B⃗ .ds=ϕenclosed —–(1)

Scalar Electric Flux (𝜓) Scalar Magnetic Flux (𝜙)

They are the imaginary lines of force radiating in an outward direction They are the circular magnetic field generated around a current-carrying conductor.

A charge can be source or sink No source/sink

Hence we can conclude that magnetic flux cannot be enclosed within a closed surface of any shape.

∯ B⃗ .ds=0 ———(2)

Applying Gauss divergence theorem to equation (2) we can convert it(surface integral)

into volume integral by taking the divergence of the same vector.

⇒∯ B⃗ .ds=∭▽.B⃗ dv ——–(3)

Substituting equation (3) in (2) we get-

∭▽.B⃗ dv=0 ——-(4)

Here to satisfy the above equation either ∭dv=0 or ▽.B⃗ =0. The volume of any body/object can never be zero.

Thus, we arrive at Maxwell’s second equation.

▽.B⃗ =0

Where,

B⃗ =μH¯ is the flux density.

⇒▽.H⃗ =0 [solonoidal vector is obtained when the divergence of a vector is zero. Irrotational vector is obtained when the cross product is zero]

Maxwell Third Equation

Statement: Time-varying magnetic field will always produce an electric field.

Maxwell’s 3rd equation is derived from Faraday’s laws of Electromagnetic Induction. It states that “Whenever there are n-turns of conducting coil in a closed path which is placed in a time-varying magnetic field, an alternating electromotive force gets induced in each and every coil.” This is given by Lenz’s law. Which states that ”An induced electromotive force always opposes the time-varying magnetic flux.”

When two coils with N number of turns; A primary coil and Secondary coil. The primary coil is connected to an alternating current source and the secondary coil is connected in a closed loop and is placed at a small distance from the primary coil. When an AC current passes through the primary coil, an alternating electromotive force gets induced in the secondary coil. See the figure below.

Mathematically it is expressed as –

Alternating emf, emfalt=−Ndϕdt ——–(1)

Where,

N is the number of turns in a coil

𝜙 is the scalar magnetic flux.

The negative sign indicates that the induced emf always opposes the time-varying magnetic flux.

Let N=1,

⇒emfalt=−dϕdt ——-(2)

Here, the scalar magnetic flux can be replaced by –

ϕ=∬B⃗ .ds⃗ ——–(3)

Substitute equation (3) in (2)

emfalt=−ddt∬B⃗ .ds⃗

Which is an partial differential equation given by-

emfalt=∬−δB⃗ δt.ds⃗ ———–(4)

The alternating electromotive force induced in a coil is basically a closed path

⇒emfalt=∮E⃗ .dl⃗ ——-(5)

Substituting equation (5) in (4) we get-

⇒∮E⃗ dl⃗ =∬−δB⃗ δt.ds⃗ —–(6)

The closed line integral can be converted into surface integral using Stoke’s theorem. Which states that “Closed line integral of any vector field is always equal to the surface integral of the curl of the same vector field”

⇒∮E⃗ dl⃗ =∬(▽×E⃗ ).ds⃗ ———(7)

Substituting equation (7) in (6) we get-

⇒∬(▽×E⃗ )ds⃗ =∬−δB⃗ δt.ds⃗ ———(8)

The surface integral can be canceled on both sides. Thus, we arrive at Maxwell’s third equation

▽×E⃗ =−δB⃗ δt

Hence, we can conclude that the time-varying magnetic field will always produce an electric field.

Extended Maxwell’s third equation or Maxwell’s third equation for the static magnetic field

Which states that Static electric field vector is an irrotational vector.

Static field implies the time-varying magnetic field is zero,

⇒−δB⃗ δt=0 ⇒▽×E⃗ =0

Maxwell’s Fourth Equation

It is based on Ampere’s circuit law. To understand Maxwell’s fourth equation it is crucial to understand Ampere’s circuit law,

Consider a wire of current-carrying conductor with the current I, since there is an electric field there has to be a magnetic field vector around it. Ampere’s circuit law states that “The closed line integral of magnetic field vector is always equal to the total amount of scalar electric field enclosed within the path of any shape” which means the current flowing along the wire(which is a scalar quantity) is equal to the magnetic field vector (which is a vector quantity)

Mathematically it can be written as –

∮H⃗ .dl⃗ =Ienclosed—–(1)

Any closed path of any shape or size will occupy one surface area. Thus, L.H.S of equation (1) can be converted into surface integral using Stoke’s theorem, Which states that “Closed line integral of any vector field is always equal to the surface integral of the curl of the same vector field”

∮H⃗ .dl⃗ =∬(▽×H⃗ ).ds⃗ —-(2)

Substituting equation(2) in (1) we get-

∬(▽×H⃗ ).dl⃗ =Ienclosed ——(3)

Here, ∬(▽×H⃗ ).dl⃗ is a vector quantity and Ienclosed is a scalar quantity.

To convert this scalar quantity into the vector, multiply Ienclosed by current density vector J⃗ . That is defined by scalar current flowing per unit surface area.

J⃗ =Isa^N measured using (A/m2)

Therefore,

J⃗ =Differenceinscalarelectricfielddifferenceinvectorsurfacearea J⃗ =dIds dI=J⃗ .ds⃗ ⇒I=∬J⃗ .ds⃗ ——-(4)

Thus, the scalar quantity is converted into vector quantity. Substituting equation(4) into (3) we get-

⇒∬(▽×H⃗ ).dl⃗ =∬J⃗ .ds⃗ ———(5)

In the above equation, R.H.S and L.H.S both contains surface integral. Hence we can cancel it.

Thus, we arrive at Maxwell’s fourth equation-\

J⃗ =▽×H⃗ ——(6)

We can conclude that current density vector is curl of static magnetic field vector.

On applying time-varying field(differentiating by time) we get-

▽×J⃗ =δρvδt ———(7)

Apply divergence on both sides of equation(6)-

▽.(▽×H⃗ )=▽×J⃗

The divergence of curl of any vector will always be zero.

▽.(▽×H⃗ )=0 ——-(8)

Thus, from equation (7) and(8) we can write that-

δρvδt=0

Which contradicts the continuity equation for the time-varying fields.

To overcome this drawback we add a general vector to static field equation(6)

(▽×H⃗ )=J⃗ +G⃗ ——(9)

Applying divergence on both the sides-

▽.(▽×H⃗ )=▽.(J⃗ +G⃗ )

The divergence of the curl of any vector will always be zero.

0=▽.J¯+▽.G⃗ ▽.G⃗ =−▽.J⃗ —–(10)

Substituting equation(6) in (10) we get-

▽.G⃗ =δρvδt —–(11)

By Maxwell’s first equation,

ρv=▽.D⃗

Substituting the value of ρv in equation (11) we get-

▽.G⃗ =δ(▽.D⃗ )δt —–(12)

Here, δδt is time varient and ▽.D⃗ is space verient and both are independent to each other. Thus, on rearranging equation (12) we get-

▽.G⃗ =▽.δ(D⃗ )δt

Thus canceling the like terms we get-

G⃗ =δD⃗ δt=J⃗ D —–(13)

Substituting them in (▽×H⃗ )=J⃗ +G⃗

This is a insulating current flowing in the dielectric medium between two conductors.

Hence maxwell’s forth equation will be

⇒(▽×H⃗ )=J⃗ +J⃗ D

Or

⇒(▽×H⃗ )=J⃗ +δD⃗ δt

Where, J⃗ D is Displacement current density.

We know that magnetic flux is equal to product of the electric flux and permittivity.

D⃗ =εE⃗ —–(14)

Substituting equation (14) in (13) we get-

δ(εE⃗ )δt=J⃗ D ⇒J⃗ D=εδ(E⃗

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