## Maxwell’s Equations (Integral Form)

It is sometimes easier to understand Maxwell’s equations in their integral form; the version we outlined last time is the differential form.

For Gauss’ law and Gauss’ law for magnetism, we’ve actually already done this. First, we write them in differential form:

We pick any region we want and integrate both sides of each equation over that region:

On the left-hand sides we can use the divergence theorem, while the right sides can simply be evaluated:

where is the total charge contained within the region . Gauss’ law tells us that the flux of the electric field out through a closed surface is (basically) equal to the charge contained inside the surface, while Gauss’ law for magnetism tells us that there is no such thing as a magnetic charge.

Faraday’s law was basically given to us in integral form, but we can get it back from the differential form:

We pick any surface and integrate the flux of both sides through it:

On the left we can use Stokes’ theorem, while on the right we can pull the derivative outside the integral:

where is the flux of the magnetic field through the surface . Faraday’s law tells us that a changing magnetic field induces a current around a circuit.

A similar analysis helps with Ampère’s law:

We pick a surface and integrate:

Then we simplify each side.

where is the flux of the electric field through the surface , and is the total current flowing through the surface . Ampère’s law tells us that a flowing current induces a magnetic field around the current, and Maxwell’s correction tells us that a changing electric field behaves just like a current made of moving charges.

We collect these together into the integral form of Maxwell’s equations:

[...] law, we’re already done, since it’s exactly the third of Maxwell’s equations in integral form. So far, so [...]

Pingback by Deriving Physics from Maxwell’s Equations « The Unapologetic Mathematician | February 3, 2012 |

who is maxwell and why did he decide to come with his equations that are difficult to understand

Comment by TPL | May 17, 2012 |

Blame the universe’s electromagnetism for being most easily calculable with vector calculus.

Comment by Jose | February 12, 2013 |

That note is clear and good prapared

Comment by Habamenshi Pierre Claver | November 23, 2012 |

why are these equations called maxwell’s equations while none of them are derived or proved by mexwell ,all of them were alread present

Comment by abdur rahim | December 26, 2012 |

where on earth is maxwell from?

Comment by Uc | June 20, 2013 |