Maxwell's Equations: Help to truly understand them

[TFM]

Homework Statement

Help to fully appreciate Maxwell's Equations

Homework Equations

Maxwell's Equations (Differential Form):

$$\nabla \cdot \vec{E} = \frac{\rho}{\epsilon_0}$$
$$\nabla \cdot \vec{B} = 0$$
$$\nabla \times \vec{E} = \frac{\partial \vec{B}}{\partial t}$$
$$\nabla \times \vec{B} = \mu_0 J + \mu_0\epsilon_0\frac{\partial \vec{E}}{\partial t}$$


Maxwell's Equations (Integral Form):

$$\oint_{closed surface} \vec{E} \cdot dS = \frac{1}{\epsilon_0}\int_{volume} \rho dv$$
$$\oint_{closed surface} \vec{B} \cdot dS = 0$$
$$\oint_{loop} \vec{E} \cdot dl = -\frac{d\Phi}{dt}$$
$$\oint_{loop} \vec{B} \cdot dl = \mu_0I + \mu_0\epsilon_0 \int_{surface}\frac{\partial E}{\partial t} \cdot ds$$

The Attempt at a Solution

Okay. So this technically isn't a Homework Question as such. In another thread, I was told that to help with some of the more complicated parts of Electromagnetism, I need to fully appreciate what the original Maxwell Equations actually mean/represent.

Could anyone hare help me with this, Please?

Thanks,

TFM

 

[Defennder]

Don't you have a textbook or something? A complete explanation, or even a sufficient explanation would almost certainly require a lot of background explanation and knowledge. It's hard to do that here.

 

[TFM]

I have 'Introduction to Electrodynamics' by David Griffiths

 

[Defennder]

You still haven't asked any specific questions.

 

[TFM]

Well firstly, I'll put a link to the aforementioned thread:

https://www.physicsforums.com/showthread.php?t=276223

So how do Maxwell's Laws really relate to Electromagnetic waves? I know the speed of light (in a vacuum) is given from them by:

$$c = \frac{1}{sqrt{\epsilon_0 \mu_0}}$$$$c = \frac{1}{\sqrt{\epsilon_0 \mu_0}}$$

But how else do they relate to EM Waves?

TFM

 

[physics girl phd]

Firstly... you can finesse through some mathematics with Maxwells equation and derive the "wave equation" -- which is a standard partial differential equation. This derivation is available EVERYWHERE, so you can look it up (I see no reason to slog through it here). Interesting enough perhaps in free space (because of the vector nature of the periodic propagating field disturbances... the exponential respentations of which you show in your link), solutions to the wave equation are even more interesting when you talk about "boundary value problems" ... things like reflection and refraction, propagating modes in restricted geometries (waveguides), interesting effects in materials (like second harmonic generation) etc. Maxwell's equations not only provide the wave equation, but they also restrict the boundary conditions in certain ways.

Therefore, to truly understand Maxwell's Equations, I suggest you take some form of a PDE / Boundary Value Problem course, or at least read up on such (there are other neat BVP's like heat flow equations... and the wave equation also can be related to sound waves: via vibrations on 1-D strings, d2-D drumheads, and in 3-D pipes). The text I used when I took such a course was an early edition of David Power's Boundary Value Problems. I was fortunate enough to have taken it BEFORE I took an upper-level undergraduate E&M course... I'd say it helped me enormously, which is why I've always personally been fond of EM, despite other's qualms (fellow grad students dreaded when we encountered the standard rite of passage in Jackson's Electrodynamics in grad school... I found it simply routine fun. )

 

[TFM]

So the Wave equations from Maxwell's eqn would be when you curl of the curl E/B and get, respectively:

$$\nabla^2\vec{E} = \mu_0\epsilon_0 \frac{\partial^2\vec{E}}{\partial t^2}$$

$$\nabla^2\vec{B} = \mu_0\epsilon_0 \frac{\partial^2\vec{B}}{\partial t^2}$$

and since nabla^2 B gives:

$$\nabla^2 B = \frac{\partial^2\vec{B_x}}{\partial x^2} + \frac{\partial^2\vec{B_y}}{\partial y^2} + \frac{\partial^2\vec{B_z}}{\partial z^2}$$

Edit, error in Latex above should be:
$$\nabla^2 B = \frac{\partial^2\vec{B_x}}{\partial x^2} + \frac{\partial^2\vec{B_y}}{\partial y^2} + \frac{\partial^2\vec{B_z}}{\partial z^2}$$and similarly for E:

$$\nabla^E = \frac{\partial^2\vec{E_x}}{\partial x^2} + \frac{\partial^2\vec{E_y}}{\partial y^2} + \frac{\partial^2\vec{E_z}}{\partial z^2}$$

Edit, error in Latex above should be:
$$\nabla^2 E = \frac{\partial^2\vec{E_x}}{\partial x^2} + \frac{\partial^2\vec{E_y}}{\partial y^2} + \frac{\partial^2\vec{E_z}}{\partial z^2}$$and so:

$$\frac{\partial^2\vec{B_x}}{\partial x^2} + \frac{\partial^2\vec{B_y}}{\partial y^2} + \frac{\partial^2\vec{B_z}}{\partial z^2} = \mu_0\epsilon_0 \frac{\partial^2\vec{B}}{\partial t^2}$$

and

$$\nabla^2\vec{E} = \mu_0\epsilon_0 \frac{\partial^2\vec{E}}{\partial t^2}$$$$\nabla^2\vec{B} = \mu_0\epsilon_0 \frac{\partial^2\vec{B}}{\partial t^2}$$

and since nabla^2 B gives:

$$\nabla^B = \frac{\partial^2\vec{B_x}}{\partial x^2} + \frac{\partial^2\vec{B_y}}{\partial y^2} + \frac{\partial^2\vec{B_z}}{\partial z^2}$$

and similarly for E:$$\frac{\partial^2\vec{E_x}}{\partial x^2} + \frac{\partial^2\vec{E_y}}{\partial y^2} + \frac{\partial^2\vec{E_z}}{\partial z^2} = \mu_0\epsilon_0 \frac{\partial^2\vec{B}}{\partial t^2}$$

TFM

Edit: I'm not sure what went wrong above, but the order is slightly off. The final Equations should be for E:$$\frac{\partial^2\vec{E_x}}{\partial x^2} + \frac{\partial^2\vec{E_y}}{\partial y^2} + \frac{\partial^2\vec{E_z}}{\partial z^2} = \mu_0\epsilon_0 \frac{\partial^2\vec{B}}{\partial t^2}$$

and B:$$\frac{\partial^2\vec{B_x}}{\partial x^2} + \frac{\partial^2\vec{B_y}}{\partial y^2} + \frac{\partial^2\vec{B_z}}{\partial z^2} = \mu_0\epsilon_0 \frac{\partial^2\vec{E}}{\partial t^2}$$