Maxwell’s Equations: Derivations & Integral Forms

Maxwell's four equations are a set of coupled partial differential equations in electromagnetism. Maxwell’s Equations were derived by James Clerk Maxwell who explained the behavior of electric and magnetic fields, their interactions and the influence of objects. With Lorentz Force Law, they form the foundation of classical electromagnetism, classical optics, and electric circuits. 

The four Maxwell equations can be expressed as:

• div D = ρ
• div B = 0
• curl E = -dB/dt
• curl H = dD/dt + J.

Maxwell’s equation illustrated the speed of electromagnetic waves is the same as the speed of light. This is used in understanding the principle of antennas. The flow of electric current produces a magnetic field. When the flow of charges varies with time, it induces an electric field. The separated positive and negative charges give rise to an electric field that propagates a magnetic field when it varies with time. 

Read More: Electrostatics

Key Terms: Maxwell’s Equation, Gauss’s Law of Electricity, Gauss’s Law of Magnetism, Faraday’s Law of Induction, Ampere’s Law, Electromagnetic Induction, Electric Field, Electric Charges

---

Maxwell's Four Equations

[Click Here for Sample Questions]

Maxwell’s four equations are a combination of the following four laws from Current Electricity. They contribute to the understanding of a specific aspect of electromagnetism.

• Gauss’s Law of Electricity: Describes how electric charges produce electric fields.
• Ampere’s Law of Current in a Conductor: Relates magnetic fields to electric current.
• Faraday’s Law of Electromagnetic Induction: Explains how a changing magnetic field induces an electric current.
• Gauss’s Law of Magnetism Specifies that magnetic monopoles do not exist, and magnetic field lines are always closed loops.

Maxwell derived a set of four equations that formed the very base of electric circuits. His equations explain the working of static electricity, electric current, Power generation, electric motor, lenses, radio technology etc. From Maxwell’s equations, it can be concluded that in an electromagnetic wave, the electric and magnetic fields are perpendicular to each other and also to the direction of propagation.

Maxwell’s Equations

In the mid-19th century, James Clerk Maxwell transformed our understanding of electromagnetism by unifying the laws of electricity and magnetism. His four equations not only united these forces but also predicted the existence of electromagnetic waves, the basis for technologies like radio. Maxwell's work influenced quantum mechanics and relativity, shaping 20th-century physics. 

Check Out:

Maxwell’s Equations Detailed Video Explanation:

Also Read: Charging by Induction

---

Derivations of Maxwell’s Equations

[Click Here for Previous Year Questions]

The laws given below are useful in the derivation of Maxwell’s equations, which describe the working of the electric fields that can create magnetic fields and vice-versa:

---

Maxwell’s First Equation

[Click Here for Sample Questions]

Maxwell’s First equation is derived from Gauss's Law of Electricity, stating that:

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 a positive charge to a 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.

Maxwell’s First Equation Derivation

Mathematically derivation of Maxwell’s first equation can be derived as:

Maxwells Equations – Closed Surface with Enclosed Charge

For a closed system, the enclosed charge is the product of the surface integral and the electric flux density.

It can be mathematically represented as:

∯ \(\overrightarrow{D}.d\overrightarrow{s}= Q_{enclosed}\) ---- (1)

Closed systems have only volumes so converting surface integrals to volume integrals by using divergence of vectors:

∯ \(\overrightarrow{D}.d\overrightarrow{s}= \iiint \Delta.\overrightarrow{D} d \overrightarrow{v}\) ---(2)

Combining equations (1) and (2) we get

\(\iiint \Delta.\overrightarrow{D} d \overrightarrow{v}= Q_{enclosed}\) ---- (3)

Charges get redistributed over the volume, thus the volume change ρ can be represented as

\(\rho v dv= \frac{dQ}{dv}\) measured in C/m³

Rearranging we get

\(dQ= \rho v dv\)

\(Q= \iiint \rho vdv\) ---(4)

Substituting (4) in (3) we get

\(\iiint \Delta. Ddv= \iiint \rho vdv\)

Thus, Maxwell’s First Equation is ∇.Ddv=ρv 

Gauss Law

Gauss's law explains an electric field's nature around electric charges. In a closed surface, Gauss law implies that the net flux of an electric field is in direct proportion to the enclosed electric charge.

Thus, ∇.D = ρ_v 

The inverted triangle is known as the divergence operator. When the electric charge is seen to exist, the divergence of “D” at that specific point becomes non-zero. Elsewhere, it is zero.

Read More: Gaussian Surface

---

Maxwell’s Second Equation

[Click Here for Previous Year Questions]

Maxwell’s Second Equation is based on Gauss’s Law of Magnetism states that:

• 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 i.e. if field lines enter an object, they will also exit from it.
• A Gaussian surface has no total magnetic field.
• The magnetic field is known as a solenoidal vector field.

Maxwell’s Equations – Gauss's Law of Magnetism

Difference Between Scalar Electric Flux and Scalar Magnetic Flux

The differences between scalar electric flux and scalar magnetic flux are tabulated below:

Maxwell’s Second Equation Derivation

Mathematically Maxwell’s Second Equation derivation is 

∯\(\overrightarrow{B}.ds=\phi _{enclosed}\) ----- (1)

Magnetic flux cannot be enclosed inside a surface

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

Converting surface integral to a volume integral using divergence of vectors

∯ \(\overrightarrow{B}.ds=\iiint \Delta.\overrightarrow{B}dv\) ---- (3)

Substituting (3) in (2) we get, 

\(\iiint \Delta.\overrightarrow{B}dv= 0\) ---- (4)

The above equation can be satisfied using only the following two conditions:

• \(\iiint dv= 0\)
• \(\Delta.\overrightarrow{B}=0\)

However, the volume of an object cannot be 0, thus \(\Delta.\overrightarrow{B}=0\)

Here \(\overrightarrow{B}=\mu \overrightarrow{H}\), Thus, Maxwell’s Second Equation is

 \(\Delta. \overrightarrow{H}=0\) 

Gauss Magnetism Law

Gauss law of magnetism can be expressed by using Gauss law for the electric field, stating that:

∇.D = ρ_v 

∇.B = 0

Both equations exhibit the divergence of the field. The first equation claims that the divergence of the electric flux density D is equivalent to the volume of electric charge density. The second equation expresses that the divergence of the Magnetic Flux Density (B) is null.

Check Out:

---

Maxwell’s Third Equation

[Click Here for Sample Questions]

Maxwell’s third equation is based on Faraday’s Laws of Electromagnetic Induction which states that:

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. 

Maxwells Equations – Faraday’s Law of Electromagnetic Induction

• 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, and 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.

According to Lenz's Law,” An induced electromotive force opposes the time-varying magnetic flux always. This particular concept of electromagnetic induction is responsible for the basic operation of various electric devices- rotation of bar magnets for creating change in magnetic fields that further creates electric fields in a conducting wire placed near it.

Maxwell’s Third Equation Derivation

Mathematically the derivation of Maxwell’s Third Equation is 

emfalt = -N d∅dt --(1)

Here, N denotes the number of turns in a coil

∅ denotes the scalar magnetic flux

Negative sign denotes the induced emf which always opposes the time-varying magnetic flux.

If, N = 1

Then, emfalt = - d∅dt --- (2)

Here, we replace the scalar magnetic flux with

∅ = B . ds ------ (3)

Substituting the equation (3) in (2), we get

emfalt = - ddt B . ds

→ emfalt = -δBδt . ds ----- (4)

The alternate electromotive force which will be induced in a coil in closed path is

emfalt = E .dl ------- (5)

Substituting the equation (5) in (4)

E .dl = -δBδt . ds ----- (6)

Using Stokes theorem in equation (6)

→ E .dl = (∇  × E ). ds ----- (7)

Substituting again equation (7) in (6) we get,

 → (∇  × E ). ds = -δBδt . ds ----- (8)

If the surface integral gets cancelled on both sides, we get Maxwell’s Third Equation 

∇ × E = - δBδt 

Since the time-varying magnetic field is zero,

→ - δBδt = 0

 → ∇ × E  = 0

Faraday's law

As per Faraday’s Law: \(\bigtriangledown \times E = - \frac{\delta B}{\delta t}\)

According to Faraday's law, when a battery is disconnected, no amount of electricity can flow via the wire. Therefore, no magnetic flux can be induced in the iron (Magnetic Core). Iron, here, functions like a magnetic field which flows easily in a magnetic material. The primary purpose of the core is to create a path for the flow of magnetic flux. The setup can be demonstrated as:

Faraday's Law Setup

---

Maxwell’s Fourth Equation

[Click Here for Previous Year Questions]

Maxwell’s fourth equation is derived from Ampere’s Law which states that:

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. This Maxwell’s equation also defines the displacement current. The electric current and displacement current through a closed surface is directly proportional to the induced magnetic field around any closed loop. 

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

Maxwell’s Fourth Equation Derivation

Mathematical Representation of Maxwell's Equation – 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)

Cancelling 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

Ampere’s Law

Ampere’s law demonstrates the relationship between the flow of electric current and the magnetic field surrounding it. Assume that the wire carries a current I, thus the current generates a magnetic field surrounding the wire. Thus,

\(\bigtriangledown \times H = \frac{\delta D}{\delta t} + J\)

---

Handwritten Notes on Maxwell’s Equations

[Click Here for Sample Questions]

Provided below are some important handwritten notes on Maxwell’s Equations, electromagnetic wave and wave equations for quick reference. 

Derived from fundamental principles, Maxwell’s equations show the dynamics of electromagnetic phenomena. Maxwell's equations shape diverse fields from telecommunications to quantum mechanics, shaping our understanding of the physical universe. 

---

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. 
• Maxwell’s Equations have been derived from: Gauss’s Law of Electricity, Ampere’s Law of Current in a Conductor, Faraday’s Law of Electromagnetic Induction, and Gauss’s Law of Magnetism.
• Gauss's law of electricity defines the link between a static electric field and the electric charges that generate the electric field.
• Gauss's Law of magnetism states that "net magnetic flux that passes through a closed surface is 0".
• Faraday's Law of induction states that "the magnetic field generated around a closed loop is equal to the work done to move a unit charge within that loop".
• Ampere's Circuit Law states that "A magnetic field can be generated by the electric current or changing the electric field".
• Maxwells equations are based on the concept of how the magnetic fields and electric fields are generated by currents, charge and by change of fields.
• Maxwell’s four Equations can be summarised as:

Also Check:

---

Previous Year Questions

1. The system will be in equilibrium if the value of q is…. [WBJEE 2016]
2. A hollow metal sphere of radius R  is uniformly charged. The electric field due to the sphere... [NEET 2019]
3. The energy required to rotate the dipole by 90^∘ is….[NEET 2013]
4. When the  Gaussian spherical surface is doubled , then then the outward electric flux will be… [NEET 2011]
5. The equilibrium separation between the balls... [NEET 2013]
6. In the given circuit, what will be the equivalent resistance between the points…. [JIPMER 2006]
7. two charges of equal amount +Q are placed on a line...[WEBJEE 2016]
8. The total energy of a system is...[JEE Main 2018]
9. If the ammeter has a coil of resistance 480ohm and a shunt of 20ohm , the reading in the ammeter will be….[NEET 2015]
10. For a plane electromagnetic wave propagating in x direction… [NEET 2021]
11. The magnetic field in a travelling electromagnetic wave has a peak value of 20… [VITEEE 2018]
12. A 100 ohm resistance and a capacitor of 100 ohm reactance are connected in series… [NEET 2016]
13. A parallel plate capacitor of capacitance of 20 micro Farad… [NEET 2019]
14. A radiation of energy 'E ' falls normally on a perfectly reflecting surface… [NEET 2015]
15. For a transparent medium relative permeability and permittivity… [NEET 2019]