Help understanding Maxwell's Equations please

[PhysicsMan68]

I have having trouble understanding Maxwell's Equations. Can anyone recommend some good book or website that can help me to understand these Equations? How can electric and magnetic fields travel perpendicular to each other? What causes electromagnetic waves to first radiate from its source? I greatly appreciate any comments or suggestions.

 

[anuttarasammyak]

Feynmann https://www.feynmanlectures.caltech.edu/II_18.html is a classic.

 

[Vanadium 50]

To get waves, start with \nabla \times \nabla \times E and the wave equation will pop out. If what I wrote looks unfamiliar to you, you will need to brush up on your vector calculus before tacking Maxwell.

 

[sophiecentaur]

you will need to brush up on your vector calculus

There's no hand waving - 'it makes sense' way of getting to know Maxwell/s equations. I imagine the OP has done a search and is looking for a short cut. I don't know of one.

 

[Vanadium 50]

Hopefully the OP will clarify. It would be a rather odd title for a thread if he wants a math-less description of equations, though!

 

[PhysicsMan68]

What is a good book for learning vector calculus? I do have a book but it doesn't seem to be a good one.

 

[Orodruin]

What is a good book for learning vector calculus? I do have a book but it doesn't seem to be a good one.

There are a lot of books covering introductory vector calculus. If you have one and you are not happy you need to provide the reference and why you are not happy with it as well as tell us what your background is. Otherwise we are just stabbing in the dark.

 

[Frabjous]

Marsden and Tromba is a standard reference.

Given your EM focus, you might take a look at Vector Analysis by Phillips.

 

[Sagittarius A-Star]

What causes electromagnetic waves to first radiate from its source?

In addition to what others wrote ...

Definition of the nabla operator:
$\vec \nabla := ({\partial \over \partial x},{\partial \over \partial y},{\partial \over \partial z })$

In vacuum, in the absence of sources, Maxwell's equations are
$\vec \nabla \cdot \vec E = 0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1)$
$\vec \nabla \times \vec B = \epsilon_0\mu_0{\partial \vec E \over \partial t}\ \ (2)$
$\vec \nabla \cdot \vec B = 0\ \ \ \ \ \ \ \ \ \ \ \ \ \ (3)$
$\vec \nabla \times \vec E = -{\partial \vec B \over \partial t}\ \ \ \ \ (4)$

An identity of vector calculus is
$\vec\nabla \times (\vec\nabla \times \vec v ) = \vec\nabla (\vec\nabla \cdot \vec v) - \vec\nabla^2 \vec v$.

This identity applied to the electric field and using equation (1):
$\vec\nabla \times (\vec\nabla \times \vec E ) = 0 - \vec\nabla^2 \vec E$
It follows, using equation (4):
$\vec\nabla \times (-{\partial \vec B \over \partial t} ) = - \vec\nabla^2 \vec E$
${\partial \over \partial t}(\vec \nabla \times \vec B) = \vec\nabla^2 \vec E$
It follows, using equation (2):
${\partial \over \partial t}(\epsilon_0\mu_0{\partial \vec E \over \partial t}) = \vec\nabla^2 \vec E$

Using $c^2={1 \over\epsilon_0\mu_0 }$, the result is a 2^nd order linear differential equation:

$\vec\nabla^2 \vec E - {1 \over c^2}{\partial^2 \vec E \over \partial t^2} = 0$

In an analog way, one can derive the same for the magnetic field:

$\vec\nabla^2 \vec B - {1 \over c^2}{\partial^2 \vec B \over \partial t^2} = 0$

These differential equations are solved by a function, that describes the propagation of an electromagnetic wave with speed $c$ in vacuum.

Source: Book "Covariant Electrodynamics" by John M. Charap, chapters 2.1 and 3.4
https://www.amazon.com/Covariant-Electrodynamics-John-M-Charap/dp/1421400146?tag=pfamazon01-20

 

[phyzguy]

If you start with The Feynman Lectures Volume 2, Chapter 1
and work through it chapter by chapter carefully, you will get there. He explains the vector calculus, assuming you know basic calculus.

 

[jtbell]

If you're not familiar with the online edition of the Feynman lectures... to move from one chapter to the next, click on the right-pointing triangle in the third row of the palette at the top right of the page. Chapter 2 is where the vector calculus begins.

In Griffiths's Introduction to Electrodynamics, the first chapter or two contain a review of vector calculus. It might be sufficient as an introduction for someone who knows single-variable calculus.

 

[Meir Achuz]

Try "Understanding Vector Calculus: Practical Development and Solved Problems".