Curl of the curl of E or B field

[DoobleD]

A modern standard way of deriving the EM wave equation from Maxwell's equations seems to be by taking the curl of curl of E and B field respectively, and use some vector identity. See for instance on wikipedia.

So, I have a basic understanding of the curl of a vector field. Defined as the closed loop line integral divided by the infinitesimal area it encloses, curl is a vector field itself, offering some "rotation related" information of the vector field from which it is derived.

For a velocity field, the curl of the field at a point would be proportional to the field angular velocity at this point. For a force field, it would be proportional to the angular acceleration of the field at this point.

So far so good. Now, what means taking the curl of the curl of a force field ? Why is this justified when deriving EM waves ? Let's say you never looked at the EM wave equation derivation, and you just know what the curl of a field is. You stare at Maxwell's equation in their differential form, in vacuum.

Why would you take the curl of the curl of the fields ? What's the reason behind that move ? Can't just be a random idea.

The only logical interpretation I can see is that the curl of the curl of say the E field, is the rate at which the angular acceleration of the field changes at the given point. But that doesn't tell me why one would be interested with that.

 

[Dale]

Why would you take the curl of the curl of the fields ? What's the reason behind that move ? Can't just be a random idea.

Why can't it just be a random idea? Regardless of the motivation, the analysis is valid.

Often they call this an ansatz, which is basically an educated guess. It is a standard and accepted approach to math.

 

[DoobleD]

Why can't it just be a random idea? Regardless of the motivation, the analysis is valid.

Often they call this an ansatz, which is basically an educated guess. It is a standard and accepted approach to math.

Well that's true it can actually be a random try. I just thought there was a particular reason or clue I wasn't aware of. Thanks for answering.

 

[Vanadium 50]

It can be random, but in this case it's not entirely random. If you want a wave equation, you need another derivative. If you want a vector wave equation, you need a vector operator. So curl is a good starting point.

 

[DoobleD]

If you want a wave equation, you need another derivative. If you want a vector wave equation, you need a vector operator.

Makes sense.

 

[Chandra Prayaga]

The two curl equations in Maxwell equations are coupled first order partial differential equations. So you can wonder if we can decouple them and write second order PDEs for each of the fields, by using the standard method of differentiating one of them again and then substituting from the other. For a more elementary example, for a charged particle in a magnetic field, moving in the xy plane, with the magnetic field along the z-axis:

dv/dt = (q/m) (v x B)

This is two first order coupled differential equations:

dv_x/dt = (q/m) v_y B (1)
dv_y/dt = - (q/m) v_xB (2)

To decouple these, differentiate (1) again wrt t, giving the first derivative of v_y on the right. Now substitute for this from (2), and that gives you a de-coupled second order differential equ. for v_x.

 

[DoobleD]

The two curl equations in Maxwell equations are coupled first order partial differential equations. So you can wonder if we can decouple them and write second order PDEs for each of the fields, by using the standard method of differentiating one of them again and then substituting from the other. For a more elementary example, for a charged particle in a magnetic field, moving in the xy plane, with the magnetic field along the z-axis:

dv/dt = (q/m) (v x B)

This is two first order coupled differential equations:

dv_x/dt = (q/m) v_y B (1)
dv_y/dt = - (q/m) v_xB (2)

To decouple these, differentiate (1) again wrt t, giving the first derivative of v_y on the right. Now substitute for this from (2), and that gives you a de-coupled second order differential equ. for v_x.

Nice! At some point I was thinking one could maybe think using the curl of curl in order to "combine" Maxwell's equations to get equations with a single unknown (E or B). But I couldn't express this idea with enough formalism. You did just that, thank you. :D