Maxwell’s Equation In Integral and Differential Form

A PRESENTATION BY: ABHIJEET GUPTA-14011011101 PRERAK TRIVEDI-140110111045 KUNAL SHETH-140110111054

CONTENTS1. Introduction2. Gauss Law for Electric Fields3. Gauss Law for Magnetic Fields4. Faraday’s Law5. The Ampere-Maxwell Law

• The equations describing the relations between changing electric and magnetic fields are known as Maxwell’s equations.

• Maxwell’s equation are extensions of the known work of Gauss, Faraday and Ampere. There are two forms of each Maxwell equation namely Integral form and Differential form(point form).

• Maxwell’s equation in the Integral form governs the interdependence of certain field and source quantities(charge and current) associated with regions in space, surfaces and volumes.

• The Differential form of Maxwell’s equations relate characteristics of the field vectors at a given point to one another and to the source densities at that point.

• The Maxwell’s equations provides the mathematical background for the study of electromagnetic waves, transmission lines and antenna.

Gauss law for Electric FieldThe total electric flux crossing the closed surface is equal to the total charge enclosed by that surface.The electric flux through the closed surface is:-

The charge can be expressed in terms of

where D is the Flux Density

Then the Gauss’s law for electric field is expressed as:-

• The previous relation is called as Integral form of Maxwell’s equation derived from Gauss’s law for the electric field.

• To relate D with del operator to convert surface integral into volume integral using Divergence theorem as:-

• Comparing two integrals, we can say that,

Gauss’s law for Magnetic Fields

The total magnetic flux crossing the closed surface is equal to zero.

The reason for this is that the magnetic flux lines are always closed in nature . Due to which a closed surface in the presence of these lines will have same number of incoming and outgoing flux lines.

The incoming flux, is considered as “ + “ and the outgoing flux, is considered as “ – “.

Faraday’s Law Michael Faraday discovered experimentally that a current was induced in a conducting loop when a magnetic flux linking the loop is changed.

The current which is induced indicates the existence of a voltage or an relationship between the induced emf and the rate of change of flux linkage is known as Faraday’s law.

In a closed path the electric potential is developed due to time varying magnetic field in the vicinity of closed path. The “ –VE “ sign in the equation indicates a LENZ LAW, which states that the current induced in a loop in such a direction as to oppose the cause producing it.

Ampere’s-Maxwell Law

Ampere's circuit law states that the line integral of the tangential component of H around a closed path is the same as the net current Ienc enclosed by the path.

So, in other words, the circulation of Magnetic Field H equals Ienc

Ampere's law is similar to Gauss's law and it is easily applied to determine H when the current distribution is symmetrical. The above equation always holds whether the current distribution is symmetrical or not but we can only use the equation to determine H when symmetrical current distribution exists. Ampere's law is a special case of Biot-Savart's law

By applying Stoke's theorem to the left-hand side of above equation, we obtain

But,

Comparing surface integrals, we get that

Application of Ampere-Maxwell’s Law

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