Maxwells Equations

Introduction

The Maxwell equations are the fundamental equations of electromagnetism, which combines Gauss’s law of electricity, Faraday's law of electromagnetic induction, Gauss’s law of magnetism and Ampere's law of current in a conductor. Maxwell's equations are a set of differential equations, which along with the Lorentz force law forms the basic foundation of electromagnetism, electric circuits and classical optics.


Maxwell's equations provide a mathematical model for static electricity, electric current, radio technologies, optics, power generation, wireless communication, radar, electric motor, lenses, etc. These equations describe the working nature of electric and magnetic fields, and how they are produced by charges, currents and due to change of electric or magnetic field.


These equations are named after a Scottish mathematical physicist James Clerk Maxwell, who formulated the classical theory of electromagnetic radiation. He published these questions by including the Lorentz Force law between the year 1861 and 1862. Maxwell's first equation proposed that ‘light is electromagnetic in nature’.


Maxwell's Equations Explained

Maxwell formulated four equations for free space, that are mentioned below:

1. First Maxwell’s Equation: Gauss’s Law for Electricity

The Gauss’s law of electricity states that, “the electric flux passing through a closed surface is equal to 1/ε0 times the net electric charge enclosed by that closed surface”.


Gauss's law of electricity describes the relationship between a static electric field and the electric charges which cause the electric field. A static electric field always points in a direction away from the positive charge, and it points in a direction towards negative charge. It also describes that the net outflow of electric field through any closed surface is directly proportional to the net amount of charge enclosed by that closed surface.


The electric field lines begin at positive charge and end at negative charge. The total number of electric field lines which passes through a closed surface, divided by the dielectric constant of free space (permittivity of vacuum), gives the total amount of charge enclosed by that closed surface.

  1. Maxwell's Equations Integral Form

e = q/e0 -------- (i)


Also, e = \[\int \overrightarrow{E} . d\overrightarrow{A}\]    ---- (ii)


Comparing equation (i) and (ii), we have:


\[\int \overrightarrow{E} . d\overrightarrow{A}\] = q/∈₀      ---- (iii)


This is the integral form of Maxwell’s 1st equation.

  1. Maxwell Equation in Differential Form

The value of total charge in terms of volume charge density is q = \[\int p dv\].


So, the equation (iii) becomes:


\[\int \overrightarrow{E} . d\overrightarrow{A} = \frac{1}{e_{0}} \int p dv\]


Applying divergence theorem on left-hand side of the above equation, we have:


\[\int (\overrightarrow{\triangledown} . \overrightarrow{E})d.V = \frac{1}{\epsilon_{0}} \int p dv\]


\[\int (\overrightarrow{\triangledown} . \overrightarrow{E})d.V - \frac{1}{\epsilon_{0}} \int p dv\] = 0


\[\int [(\overrightarrow{\triangledown} . \overrightarrow{E}) - \frac{\rho}{\epsilon_{0}}]d.V\] = 0


\[(\overrightarrow{\triangledown} . \overrightarrow{E}) - \frac{\rho}{\epsilon_{0}}\] = 0


\[(\overrightarrow{\triangledown} . \overrightarrow{E}) = \frac{\rho}{\epsilon_{0}}\]


This is the differential form of Maxwell’s 1st equation.


2. Second Maxwell’s Equation: Gauss’s Law for Magnetism

The Gauss’s law of magnetism states that “the net magnetic flux of a magnetic field passing through a closed surface is zero”. This is because magnets always occur in dipole, and magnetic monopole does not exist.


The magnetic field is generated due to the dipole nature of the magnet. The net outflow of magnetic field through any closed surface is zero. Magnetic dipoles behave like loops of current with positive and negative (i.e magnetic charges) which cannot be separated from each other.


According to the Gauss's law of magnetism, magnetic field lines make loops, and they start from the magnet and extend till infinity and back. In other words, if field lines enter an object, they will also come out of that object. The total magnetic field through a Gaussian surface is zero, and the magnetic field is a solenoidal vector field.


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This is a graphical representation of magnetic field lines which neither begin nor ends, but forms loops.


3. Third Maxwell’s Equation: Faraday’s Law of Electromagnetic Induction

Maxwell modified Faraday's law of induction. It describes the production of electric fields by a time varying magnetic field. This law describes, “ work needed for moving a unit charge around a closed loop structure equals the magnetic field transforming around that particular loop”.


The induced electric field lines are similar to that of magnetic field lines, unless they are superimposed by a static electric field. This concept of electromagnetic induction is the basic operating principle behind many electric devices like in rotating bar magnets for creating changing magnetic fields, which further produces electric fields in a nearby conducting wire.


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The Earth's magnetic field is altered in a geomagnetic storm, due to a surge in the flux of charged particles, which further induces an electric field in Earth's atmosphere.


∈ = -Ndm/dt- -------------- (v)

Since emf if related to electric field by the relation


∈ = \[\int \overrightarrow{E} . \overrightarrow{d} A\]


Also,

Putting these values in equation (v), we have:


\[\int \overrightarrow{E} . \overrightarrow{d} A\] = -N \[\int \overrightarrow{E} . \overrightarrow{d} A \int \overrightarrow{B} . \overrightarrow{d} A\]


For N = 1, we have


\[\int \overrightarrow{E} . \overrightarrow{d} A = \frac{-d}{dt} \int \overrightarrow{B} . \overrightarrow{d} A\]  ------- (vi)


This is the integral formula of Maxwell’s third equation.


Applying stoke’s theorem on L.H.S of equation (vi), we have:

\[\int (\overrightarrow{\triangledown} . \overrightarrow{E})d\overrightarrow{A} = \frac{-d}{dt} \int \overrightarrow{B} . d\overrightarrow{A}\]


\[\int (\overrightarrow{\triangledown} . \overrightarrow{E})d\overrightarrow{A} + \frac{d}{dt} \int \overrightarrow{B} . d\overrightarrow{A}\] = 0


\[(\overrightarrow{\triangledown} . \overrightarrow{E}) + \frac{d\overrightarrow{B}}{dt}\] = 0


\[(\overrightarrow{\triangledown} . \overrightarrow{E}) =  \frac{-d\overrightarrow{B}}{dt}\]


This is the differential form of Maxwell’s third equation.


4. Ampère's law with Maxwell's addition

According to Ampere’s law with Maxwell addition, “magnetic field can either be produced by electric current or by altering the electric field. The first statement is as per Ampere’s law whereas the later is according to Maxwell’s addition, the displacement current. The induced magnetic field around any closed loop is directly proportional to the electric current and the displacement current through that closed surface.


Maxwell's addition to the Ampère's establishes a relationship to make a set of equations mathematically consistent with the non-static fields, without changing the Ampère's and Gauss's laws for static fields. However, a changing electric field produces a magnetic field and vice versa. Therefore, these equations create a possibility for self-sustaining "electromagnetic waves" to travel through vacuum.


The speed of electromagnetic waves is equal to the speed of light as per the calculations and observations. Light is also a type of electromagnetic radiation (like X-rays and radio waves).


Maxwell established the relation between electromagnetic waves and light in the year 1861, from there he unified the theories of electromagnetism and optics.


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This is a magnetic core memory (1954), an application of Ampère's law. Each core stores data of the size of one bit.

FAQ (Frequently Asked Questions)

1. What is the Significance of the Maxwell Equation?

Ans - This equation explains how magnetic & electric fields are formed by variations in their current & charges. These equations describe how varying electric and magnetic fields travel at the speed of light.


Maxwell's equations are the building blocks of all modern devices like mobile phones, computers and electricity.

2. Importance of  Maxwell's Relations in Thermodynamics.

Ans - Maxwell's equations help in changing the thermodynamic variables from one set to another.


For example, suppose you want to calculate the change in entropy of a system with respect to a given pressure and at a constant enthalpy. There is no instrument to measure entropy of a system. You can however measure the temperature, pressure, and volume of a system much easily.