Maxwell’s Equations

Ananya Pandit
Introduction
Every electromagnetic form or radiation - visible light , x-ray,
sunlight that heat the earth , radio waves, televisions waves, wifi
signals, bluetooth signals, cell phone transmission and GPS all
consist solely of electric and magnetic fields. And everything we
need to know about how they propagate and interact with materials
is completely determined by Maxwell’s equation.

James Clerk Maxwell was a Scottish physicist born on June

13, 1831, and he's considered one of the most influential scientists
in the field of electromagnetism. Maxwell made significant
contributions to various areas of physics, including optics and
kinetic theory of gases, but he is most renowned for his work on
electromagnetism.
 Maxwell’s Equation
Maxwell's equations are a set of four fundamental equations in
classical electromagnetism. They describe the behavior of electric
and magnetic fields and their interaction with matter. By studying
Maxwell's equations, we gain insights into how electric and
magnetic fields are generated, how they propagate through space,
and how they interact with one another and with charged particles.
Maxwell's equations are the cornerstone of classical
electromagnetism and have profound implications for various fields
of science and technology.
Four Equations are:-
 Gauss law for
∯𝐷.𝑑𝑠=𝑄𝑒𝑛𝑐𝑙𝑜𝑠𝑒𝑑 ---- (1)
electric field Closed systems have only volumes so converting
Gauss's Law for Electric Fields is one of Maxwell's surface integrals to volume integrals by using
equations and a fundamental principle in divergence of vectors:
electromagnetism. It describes the relationship between
∯D.ds=∭Δ.D dv ---(2)
electric fields and electric charges. Here's an explanation
of Gauss's Law for Electric Fields. Combining equations (1) and (2) we get
Gauss's Law for Electric Fields states that the electric flux ∭Δ.𝐷𝑑𝑣→=𝑄𝑒𝑛𝑐𝑙𝑜𝑠𝑒𝑑 ---- (3)
through any closed surface is equal to the total charge
Rearranging we get
enclosed by that surface divided by the permittivity of
free space. dQ=ρdv
𝑄=∭𝜌d𝑣 ---(4)
Substituting (4) in (3) we get
Relation ∭Δ.Ddv=∭ρdv
Thus, Maxwell’s First Equation is
D=ε.E , where ε reduses is a ∇.D=ρ
scaler quantity in general
medium.
 ∯𝐵.𝑑𝑠=𝜙𝑒𝑛𝑐𝑙𝑜𝑠𝑒𝑑 ----- (1)
Gauss law for Magnetic flux cannot be enclosed inside a
surface
magnetic field ∯ 𝐵.𝑑𝑠=0 ----- (2)
Converting surface integral to a volume integral
using divergence of vectors
∯ B.ds=∭Δ.Bdv ---- (3)
Maxwell's second equation, also known Substituting (3) in (2) we get,
as Gauss's law for magnetism, states ∭Δ.Bdv=0 ---- (4)
that the divergence of the magnetic The above equation can be satisfied using only
field (B) is always zero:
the following two conditions:
∇⋅B=0
• ∭dv=0
This equation implies that there are no
• Δ.B=0
magnetic monopoles, and magnetic
However, the volume of an object cannot be 0,
field lines always form closed loops.
thus Δ.B=0
Faraday’s Law

Maxwell's third equation is actually related to

Faraday's law of electromagnetic induction and is Using the definition of magnetic flux (Φ=∫𝐵⋅𝑑𝐴), we can write Faraday's law
expressed as: as:
∇×𝐸=−∂𝐵/∂𝑡​ E=−d/dt​∫B⋅dA
This equation describes how a changing magnetic By applying Stokes' theorem, which relates a line integral around a closed loop
field induces an electric field. Let's derive to a surface integral over the region bounded by the loop, we have:
Maxwell's third equation from Faraday's law of 𝐸=−∫(∇×𝐵)⋅𝑑𝐴
electromagnetic induction: Now, according to Faraday's law, the induced emf is related to the curl of the
Faraday's law of electromagnetic induction states electric field:
that the electromagnetic force (emf) induced in a E=∮E⋅dl
closed loop is equal to the negative rate of change Equating the two expressions for E, we get:
of the magnetic flux through the loop: −∫(∇×𝐵)⋅𝑑𝐴=∮𝐸⋅𝑑𝑙
E=-dΦ/dt​ Using Stokes' theorem again to convert the line integral to a surface integral, we
Where: have:
• 𝐸 is the emf induced in the loop. ∫(∇×𝐸)⋅𝑑𝐴=−𝑑/𝑑𝑡∫𝐵⋅𝑑𝐴
• Φ is the magnetic flux through the loop. This equation holds true for any closed surface A, so we can equate the
integrands:
∇×E=−∂t/∂B​
 Ampere’s Maxwell Law Let's consider Ampère's law with Maxwell's addition:
Maxwell’s equation also defines the displacement ∇×𝐻=𝐽+∂𝐷/∂𝑡​
current. The electric current and displacement current Taking the divergence of both sides of this equation:
through a closed surface is directly proportional to the ∇⋅(∇×𝐻)=∇⋅𝐽+∇⋅(∂𝐷/∂𝑡)
induced magnetic field around any closed loop. Using the vector calculus identity ∇⋅(∇×H)=0, where 𝐴 is any vector
Maxwell added the displacement current to field, the left-hand side simplifies to zero:
Ampere's Law, ∇⋅𝐽+∇⋅(∂𝐷/∂𝑡)=0
• The induced magnetic field around a closed loop ∝
Rearranging terms, we get:
The electric current through the closed surface
∇⋅𝐻=𝐽+∂𝐷/∂𝑡​
• The induced magnetic field around a closed loop ∝
Thus, we have proven the equation ∇⋅𝐻=𝐽+∂𝐷/∂𝑡 which relates the
The displacement through the closed surface
divergence of the magnetic field intensity H to the electric current
density J and the time derivative of the electric displacement field D.
Conclusion
Maxwell's equations are the foundation of classical
electromagnetism and have immense importance in
physics and engineering.

Maxwell's equations revolutionized our understanding of

electromagnetism, enabling the development of numerous
technologies and providing a cornerstone for modern
physics and engineering. Their applications are vast and
diverse, spanning from everyday technologies to cutting-
edge research in fields like quantum physics and
photonics.
Thank you