M2. Gauss's Law for Magnetic Fields:

2.1 The integral form of Gauss's Law:

$\oint_{S}\;\overrightarrow{B}\;\circ\;\widehat{n}\;da = 0$  ⇒ Gauss's Law for Magnetic Field

The left side of this equation is a mathematical description of the flux of a vector field through a closed surface. In this case, Gauss's law refers to magnetic flux - the number of magnetic field lines passing through a closed surface 'S'. The right side is identically zero.

To understand the meaning of each symbol, here's an expanded view:

2.1 The integral form of Gauss's Law

Gauss's law in integral form: The total magnetic flux passing through any closed surface is Zero.

In another words: If you have a real or imaginary closed surface of any size or shape, the total magnetic flux through that surface must be zero.

Note: This does not mean that zero magnetic field lines penetrates the surface - it means that for magnetic field line that enters the volume enclosed by the surface, there must be a magnetic field line leaving that volume. Thus the inward (negative) magnetic flux must be exactly balanced by the outward (positive) magnetic flux.

2.2 The differential form of Gauss's Law: 

$\nabla\;\circ\overrightarrow{B} = 0$ ⇒ Gauss's Law for Magnetic Field

The left side of this equation is a mathematical description of the divergence of the magnetic field - the tendency of the magnetic field to "flow" more strongly away from a point than toward it, while the right side is simply zero. 

To understand the meaning of each symbol, here's an expanded view:

2.2 The differential form of Gauss's Law

Gauss's Law in differential form: The divergence of the magnetic field at any point is zero.

In other words: Why this is true is by analogy with the electric field, for which divergence at any location is proportional to the electric charge density at that locations. Since it is not possible to isolate magnetic poles, you can't have a north pole without a south pole, and the "magnetic charge density" must be zero everywhere. This means that the divergence of the magnetic field must also be zero.

We will discuss about "Faraday's Law" later on.

M1. Gauss's Law for Electric Fields:

 

1.1 The integral form of Gauss's Law:

$\oint_{S}\;\vec{E}\;\circ\;\widehat{n}\;da = \frac{q_{enc}}{\varepsilon_0}$  ⇒ Gauss's Law for Electric Fields 

The left side of this equation is a mathematical description of the electric flux - the number of electric field lines passing through a closed surface S, where as the right side is the total amount of charge contained within that surface divided by a constant called the permittivity of free space.

To understand the meaning of each symbol, here's an expanded view:

1.1 The integral form of Gauss's Law

Gauss's Law in integral form: Electric charge produces an electric field and the flux of that field passing through any closed surface is proportional to the total charge contained within that surface. 

In other words: If you have a real & imaginary closed surface of any size & shape, and there is no charge inside the surface - the electric flux through the surface must be zero. If you were to place some positive (+ve)charge anywhere inside the surface - the electric flux through the surface would be positive. If you then added an equal amount of negative charge inside the surface (making the total enclosed charge zero), the electric flux would again be zero. Remember that it is the net charge enclosed by the surface that matters in Gauss's law.

1.2 The differential form of Gauss's Law:

$\nabla\;\circ\;E = \frac{\rho }{\varepsilon_0}$ ⇒ Gauss's Law for Electric Fields

The left side of this equation is a mathematical description of the divergence of the electric field - the tendency of the field to "flow" away from a specified location, and the right side is the electric charge density divided by the permittivity of the free space. 

To understand the meaning of each symbol, here's an expanded view:

1.2 The differential form of Gauss's Law

Gauss's Law in differential form: The  electric field produced by electric charge diverges from positive charge & converges upon negative charge.

In other words: The only places at which the divergence of the electric field is not zero are those locations at which charge is present. If positive (+ve) charge is present, the divergence is positive - meaning that electric field tends to "flow" away from that location. If negative charge is present, the divergence is negative - the electric field lines tend to "flow" towards that points.

 In next note, we will discuss about "Gauss's Law for Magnetic Fields"

Maxwell's Equations:

"The most important equations of all time."

"In every branch of knowledge the progress is proportional to the amount of facts on which to build, & therefore to the facility of obtaining data."

James Clerk Maxwell

James Clerk Maxwell (1831 - 1879) was a theoretical Physicist & famous Mathematician; who took a set of known experimental laws (Faraday's Laws, Ampere's Law) & unified them into a symmetric set of equations known as Maxwell's Equation - that explains the properties of magnetic & electric field.

Maxwell was one of the first to determine the speed of propagation of electromagnetic (EM) waves was same as the speed of light. And hence he conclude that electromagnetic (EM) waves & visible light were really the same thing.

If you need to testament to the  power of Maxwell's Equations, look around you - radio, television, radar, wireless internet access & Bluetooth technology. These are the few examples of contemporary technology rooted in electromagnetic field theory - Maxwell's Equations.

Maxwell's Equations are four of the most influential equations in Physics : 

M1. Gauss's Law for Electric Fields: 

M2. Gauss's Law for Magnetic Fields: 

M3. Faraday's Law: 

M4. Ampere - Maxwell Law: