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Maxwell’s equation was used to describe what we now know as “Classical Electromagnetism”. Let us see what the laws are:
Gauss’s Law: In electromagnetism, Gauss’s Law is often called Gauss’s flux theorem, which relates the electric charges to its resulting electric field.
Gauss’s Law of Magnetism: The magnetic flux of any closed surface is always zero because of the nonexistence of isolated magnetic poles or monopoles.
Faraday’s Law: Any change in the magnetic field of the surface creates an electric field. This phenomenon is known as electromagnetic induction.
Ampere’s Law: It relates magnetism with electricity by stating the relation of magnetic field and electric current and the change of electric field that creates the magnetic field.
Integral Forms
Maxwell’s equation helps state a region of current or change in current in its integral forms.
Gauss’s Law
Gauss’s flux theorem states that the electric field resulting from a closed surface is always proportional to the electric charge it carries irrespective of the charge distribution in the closed surface. Gauss’s theorem can be used in its differential form in cases where symmetry does not mandate the uniformity of the electric field. The differential form of the theorem states that the electric field divergence is proportional to the density of charge on the surface
Gauss’s law is
ΦE = Q/ε0
where ΦE represents the electric flux (also defined as an integral or surface integral of an electric field) through a surface S of any enclosed volume V,
Q represents the total charge enclosed within the volume V, and
ε0 is the electric constant.
ΦE = E.dA
where E represents the electric field across the enclosed surface S, and
dA is the vector of the infinitesimal area of the surface S.
Gauss’s theorem is
∫s E.dA = 1/ε0 ∫Q dV
Since flux is an integral of an electric field, this expression of Gauss’s theorem is known as the integral form.
Gauss’s Law of Magnetism
Gauss’s Law of Magnetism states that as no monopole exists, the magnetic charge created across any surface or space should always sum up to 0. Magnetic dipoles can create analogous magnetic flux around a surface which has a similar mathematical formula or expression,
∫s B.dA = 0, where B is the magnetic field.
Faraday’s Law
Faraday’s theorem states that, over a closed-loop, any change in magnetic flux results in an electric field. It is termed Electromagnetic Induction. This law states that a conductor exposed to a changing magnetic field will induce a current.
The law is
∫loop E.ds = -d/dt ∫s B.dA
Ampere’s Law
Ampere’s law states that when a steady current flows through any surface, it will create a magnetic field or a flux. Any changes in the electric flux (expressed as (d/dt E. dA) will also result in changes in the magnetic field created.
The law is
∫loop B.ds = μ0∫s J.dA + μ0ε0 d/dt ∫s E.dA
Differential Forms
Differential forms of Maxwell’s equation can be used to state the implementation of the laws in individual points in space.
Gauss’s Law
In the differential form of Gauss’s law or the divergence theorem, surface integral representation for a closed surface or region is also expressed as the volume integral inside the region over a divergence.
The divergence theorem is
1/ε0 ∫∫∫ Q.dV = ∫sE.dA = ∫∫∫∇.E dV
Since the statement will hold for any closed surface, the integrands will always be equal. It can be stated as
∇. E = Q/ε0
Gauss’s Law of Magnetism
The differential form of Gauss’s Law of magnetism is the same as that of the integral form as it states that due to the nonexistence of monopoles, the magnetic charge of any closed surface should always sum up to be 0.
To understand the differential form of Faraday’s Law and Ampere’s Law, we must discuss Stoke’s theorem.
Stoke’s Theorem
Stoke’s theorem, also called Generalized Stokes Theorem, states that the line integral is related to a surface integral of a vector field. It says that the surface integral of the curl of a function over a closed surface will be equal to the line integral of any vector function over the same surface.
Faraday’s Law
The differential of Faraday’s Law is
∫loop E.ds = – d/dt ∫s B.dA
We can use Stoke’s theorem on the right-hand side of the equation to equate the integrands,
∫s ∇. E dA = – d/dt ∫s B.dA
Since the theorem holds for any closed surface, the two integrands can be realized as equal and represented as
∇. E = dB/dt
Ampere’s Law
As done for Faraday’s Law, we can invoke Stoke’s theorem in Ampere’s law by substituting the line integral of ∫B.ds in the form of the surface integral of the curl of B as
∫loop B.ds = ∫surface ∇x B dA
As Ampere’s law states,
∫loop B.ds = μ0∫s J.dA + μ0ε0 d/dt ∫s E.dA
As the surface integral can be taken for any arbitrary closed surface, thus the integrands are equal and can be expressed as
∇x B = μ0 J + μ0 ε0 dE/dt
Conclusion
Thus, Maxwell’s equation consists of four equations – Gauss’s Law, Gauss’s Law of Magnetism, Faraday’s Law, and Ampere’s law – expressed as the change in current in any region in its integral forms. When stated as a particular point in space or within the region, these equations can be represented in their differential forms. The integral form considers the whole region of current and changes in current, and the differential form considers a particular point of the surface or space.
