Maxwell's Equations: Derivations, Gauss, Faraday, Ampere’s Law

Maxwell’s Equations are a set of partial differential equations that are used to explain the nature of the working magnetic and electric fields. They are fundamental equations of electromagnetism as they explain how the electric currents and charges produce the electric and magnetic fields. The integral form of Maxwell’s equations integral explains how the electric currents and electric charges produce electric and magnetic fields. The 4 fundamental Maxwell Equations are – 

1.  ∇ D.dv = ρv
2.  ∇.\(\overrightarrow{B}\) = 0 
3. ∇ x \(\overrightarrow{E}\) = - δ\(\overrightarrow{B}\) / δt .\(\overrightarrow{ds}\) 
4. ( ∇ x \(\overrightarrow{H}\) ) = \(\overrightarrow{J}\) + \(\overrightarrow{J}\)_D

Keyterms: Electric Current, Gauss's Law, Faraday's Law, Ampere’s Law, Magnetic fields, Electric fields, Electromagnetic Induction, Electric charges, Electromagnetism

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Maxwell's Equations

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Maxwell’s Equations combine various laws like Gauss’s law of electricity, Gauss’s law of magnetism, Faraday's law of electromagnetic induction, and Ampere's law of current in a conductor.

• These equations, along with Lorentz's force, form the foundation for electromagnetism.
• They act as a model for electric current, static electricity, optics, radio technologies, power generation, etc.
• Electromagnetic waves are generated when an electric field and a magnetic field vibrate together.
• There are 4 Maxwell Equations denoted by a formula.

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Maxwell's First Equation

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Maxwell's first equation is derived from Gauss's Law of Electrostatic. Gauss's law of electricity defines the link between a static electric field and the electric charges that generate the electric field. It states that "electric flux that passes through a closed surface is 1/ε0 of the net electric charge that is enclosed in that closed surface".

A static electric field always points outwards for a positive charge, and it points inwards for a negative charge. So, we can conclude that the electric field lines travel from positive charge to negative charge. Gauss's law of electricity also states that,

• The net outflow of the electric field passing through a closed surface ∝  the net quantity of charge enclosed by that closed surface
• The total amount of charge enclosed by that closed surface = The total number of electric field lines that pass through a closed surface/the dielectric constant of free space 

Mathematical Representation of Gauss's law of electricity:

For any closed surface,

The electric flux density vector * surface integral = Charge enclosed

Let’s put this in mathematical terms, 

∯\(\overrightarrow{D}\).\(\bar{ds}\) = Q ---- eq (1)

where, \(\overrightarrow{D}\) = Electric flux density vector

\(\bar{ds}\) = Surface Integral 

Q = Enclosed Charge

A closed system can have multiple surfaces, hence the surface is not constant. But the volume of the closed system will always remain irrespective of the surface. So converting the surface integral to volume integral by taking the divergence of the surface integral vector. 

∯\(\overrightarrow{D}\).\(\overrightarrow{ds}\) = ∭ ∇ \(\overrightarrow{D}\).\(\overrightarrow{dv}\) ---- eq (2)

On using eq (1) and eq (2),

∭∇.\(\overrightarrow{D}\) \(\overrightarrow{dv}\)= Q ----- eq(3) 

Charges will be distributed over the volume of the closed surface. So, we can calculate the volume charge density by: 

ρv=dQ/dv measured in C/m³

We can rearrange the above equation as:

dQ = ρvdv

Integrating the above equation we get,

Q= ∭ρvdv ----eq (4)

Volume charge density gives the charge present within a closed surface.

Using eq (4) in eq (3), 

∭ ∇.Ddv = ∭ρvdv 

If we cancel the volume integral from both sides, we get Maxwell’s Equation (First)-

∇D.dv = ρv

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Maxwell’s Second Equation

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Maxwell’s second equation is derived from Gauss's Law of magnetism, which states that "net magnetic flux that passes through a closed surface is 0". This is due to the fact that magnets are always dipoles, and magnetic monopoles do not exist.

The dipole nature of the magnet generates a magnetic field. Through any closed surface, the net outflow of the magnetic field is zero. Magnetic dipoles act like current loops, with positive and negative charges. Gauss's law of magnetism states that the magnetic field lines generate loops, originating from the magnet to infinity & back. To put it in another way, if field lines enter an object, they will also exit from it.

Gauss's law of magnetism also states that,

• A Gaussian surface has no total magnetic field
• The magnetic field is known as a solenoidal vector field.

Gauss's law of magnetism

Mathematical Representation of Maxwell's Second Equation:

For any closed surface,

The magnetic flux density of a closed surface = Total scalar magnetic flux present on the surface (irrespective of shape and size) 

Let's quote this in mathematical terms,

∯\(\overrightarrow{B}\).ds= ϕ_enclosed ---- eq (1)

Where, 

∯\(\overrightarrow{B}\).ds = The magnetic flux density of a closed surface

ϕ_enclosed = Total scalar magnetic flux present on the surface

The magnetic flux can never be present within a closed surface of any shape. Hence, we get,

\(\overrightarrow{B}\).ds = 0 ----- eq (2) 

Converting the surface integral to volume integral by taking the divergence of the surface integral vector. 

∯\(\overrightarrow{B}\).ds = ∭ ∇.\(\overrightarrow{B}\)dv ---------- eq (3)

Using equation 3 in equation 2,

∭ ∇.\(\overrightarrow{B}\)dv = 0 ----------eq (4)

From the above equation, we can see the product is 0, so either ∭dv is 0 or ∇\(\overrightarrow{B}\) is 0. 

The volume of any object can never be 0. Therefore, we get Maxwell’s Second Equation- 

∇.\(\overrightarrow{B}\) = 0 

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Maxwell's Third Equation

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Maxwell’s third equation is derived from Faraday's Law of induction, which states that "the magnetic field generated around a closed loop is equal to the work done to move a unit charge within that loop". Electromagnetic Induction generates induced electric field lines that are similar to the magnetic field lines unless they are superimposed by an electric field that is static in nature. 

• Let us suppose we have two coils - primary and secondary coils, both having n turns and it is placed in a time-varying magnetic field.
• Connect the primary coil to an alternating current source.
• Then connect the secondary coil in a closed-loop, place it at a distance from the primary coil.
• Pass AC current through the primary coil.
• This will induce an alternating electromotive force in the secondary coil.

Faraday's Law

Mathematical Representation of Maxwell's Third equation:

Alternating electromotive force, emf:

emf_alt =−Ndϕdt ——–eq (1)

Where, 

N= number of turns in the coil

Φ = Scalar magnetic flux.

Here, the induced emf is negative because induced emf always opposes the time-varying magnetic flux

Let's suppose the coil has only 1 turn

So, N = 1

emf = −dϕ/dt

emf_alt=−dϕdt -------- eq (2)

Scalar magnetic flux, ϕ =∬\(\overrightarrow{B}\).\(\overrightarrow{ds}\) ------- eq (3) 

Using eq 3 in eq 2,

emf_alt =−d/dt ∬\(\overrightarrow{B}\).\(\overrightarrow{ds}\)

The above equation is the partial differential equation given by 

emf_alt = ∬-(δ\(\overrightarrow{B}\) / δt).\(\overrightarrow{ds}\) --------- eq (4)

The alternating emf induced in the coil is a closed path. So,

emf_alt = \(\overrightarrow{E}\).\(\overrightarrow{dl}\) ------eq (5) 

Using eq (5) in eq (4),

ϕ \(\overrightarrow{E}\).\(\overrightarrow{dl}\) = ∬- (δ\(\overrightarrow{B}\) / δt) .\(\overrightarrow{ds}\) ------eq (6) 

Stoke’s theorem states that for a vector field, the closed line integral is equal to the surface integral of the curl of that particular vector field. Therefore, converting the closed line integral into the surface integral. 

 ϕ \(\overrightarrow{E}\).\(\overrightarrow{dl}\) = ∬(∇x \(\overrightarrow{E}\) ) .\(\overrightarrow{ds}\) -------eq (7) 

Using eq (7) in eq (6),

∬(∇x \(\overrightarrow{E}\) ) .\(\overrightarrow{ds}\) = ∬- (δ\(\overrightarrow{B}\) / δt).\(\overrightarrow{ds}\) -------eq (8) 

Canceling the surface integral from both sides, we get Maxwell’s Third Equation-

∇ x \(\overrightarrow{E}\) = - δ\(\overrightarrow{B}\) / δt .\(\overrightarrow{ds}\) 

From the above derivation, we can deduce that a time-varying magnetic field always generates an electric field.

Maxwell’s third equation for the static magnetic field: 

It states that a static electric field vector is irrotational. A static field means the time-varying magnetic field will be zero.

which gives, 

- δ\(\overrightarrow{B}\) / δt = 0

∇ x \(\overrightarrow{E}\) = 0

Therefore, it's an irrotational vector.

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Maxwell's Fourth Equation

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Maxwell’s fourth equation is derived from Ampere's Circuit Law, which states that "A magnetic field can be generated by the electric current or changing the electric field". The magnetic field vector's closed line integral is equal to the total quantity of scalar electric field present in the path of that shape.

Maxwell added the displacement current to Ampere's Law, 

• The induced magnetic field around a closed loop ∝  The electric current through the closed surface
• The induced magnetic field around a closed loop ∝  The displacement through the closed surface

Ampere's Law

Mathematical Representation of Maxwell's Fourth Equation

We know, 

Closed line integral of Magnetic field vector = Total quantity of scalar electric field present

Let’s quote this in mathematical terms, 

 ϕ \(\overrightarrow{H}\).\(\overrightarrow{dl}\) = I -----eq (1) 

Stoke’s theorem states that the closed line integral of a vector field is equivalent to the surface integral of the curl of that particular vector field. Therefore, converting the closed line integral to the surface integral. 

 ϕ \(\overrightarrow{H}\).\(\overrightarrow{dl}\) = ∬(∇x \(\overrightarrow{H}\) ) .\(\overrightarrow{ds}\) ------eq (2) 

Using eq (2) in eq (1), 

∬(∇ x \(\overrightarrow{H}\) ) . \(\overrightarrow{dl}\) = I ------eq (3)

∬(∇ x\(\overrightarrow{H}\) ) .\(\overrightarrow{dl}\)= Vector Quantity

I = Scalar quantity 

To convert I into a vector, multiply I by density vector, 

\(\overrightarrow{J}\) = (I / s) âN (measured in A/m²)

\(\overrightarrow{J}\) = Difference in scalar electric field/Difference in vector electric field \(\overrightarrow{J}\)

dI /ds. dI = \(\overrightarrow{J}\).ds

I = ∬\(\overrightarrow{J}\).\(\overrightarrow{ds}\) ------- eq (4) 

Using eq (4) in eq (3), 

∬(∇ x \(\overrightarrow{H}\)).\(\overrightarrow{dl}\) = ∬\(\overrightarrow{J}\).\(\overrightarrow{ds}\) ------- eq (5)

Canceling the surface integral from both sides, we get Maxwell's Fourth equation

\(\overrightarrow{J}\) = ∇ x H

Apply time-varying field by differentiating wrt time, 

∇ x \(\overrightarrow{J}\) = δρv/ δt -------- eq (7)

Apply divergence on eq (6),

∇(∇ x \(\overrightarrow{H}\)) = ∇ x \(\overrightarrow{J}\)

We know that the divergence of the curl of any vector is zero. So, 

∇(∇ x \(\overrightarrow{H}\)) = 0 ------- eq (8)

From eq (7) and(8), 

δρv/δt =0

This contradicts the continuity equation. So to overcome this, add a general vector to eq (6)-

 (∇ x \(\overrightarrow{H}\)) = \(\overrightarrow{J}\) + \(\overrightarrow{G}\) ------- eq (9) 

Applying divergence, 

∇.(∇ x \(\overrightarrow{H}\)) = ∇.( \(\overrightarrow{J}\)+\(\overrightarrow{G}\)) 

The divergence of the curl of any vector is 0, Hence, 

0=∇.\(\overrightarrow{J}\) + ∇.\(\overrightarrow{G}\)

∇.\(\overrightarrow{G}\) = -\(\overrightarrow{J}\) -------- eq (10)

Using eq (6) in eq (10),

∇.\(\overrightarrow{G}\) = δρv/ δt --------- eq (11)

Maxwell's First equation, ρv = ∇.D

Using the value of ρv in eq (11), 

∇.\(\overrightarrow{G}\) = δ(∇.\(\overrightarrow{D}\))/ δt --------- eq (12)

Rearranging eq (12) as ∇.D is space-variant and δ/ δt is time-variant, 

∇.\(\overrightarrow{G}\) = ∇.δ(\(\overrightarrow{D}\))/ δt 

 \(\overrightarrow{G}\) = δ\(\overrightarrow{D}\))/ δt = J\(\overrightarrow{D}\) ------- eq (13)

Substituting values in ∇x \(\overrightarrow{H}\) = \(\overrightarrow{J}\) + \(\overrightarrow{G}\)

This gives the insulating current flowing between two conductors through the dielectric medium. 

Hence, the final maxwell's fourth Equation, 

( ∇ x \(\overrightarrow{H}\) ) = \(\overrightarrow{J}\) + \(\overrightarrow{J}\)_D

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Things to Remember

• Maxwell Equations are a set of partial differential equations that are used to explain the nature of the working magnetic and electric fields. 
• Gauss's law of electricity defines the link between a static electric field and the electric charges that generate the electric field.
• There are 4 Maxwell's equations explaining the nature of magnetic and electric fields.
• Maxwell’s second equation is derived from Gauss's Law of magnetism, which states that "net magnetic flux that passes through a closed surface is 0".
• Maxwell’s third equation is derived from Faraday's Law of induction, which states that "the magnetic field generated around a closed loop is equal to the work done to move a unit charge within that loop".
• Maxwell’s fourth equation is derived from Ampere's Circuit Law, which states that "A magnetic field can be generated by the electric current or changing the electric field".

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