1. Dec 20, 2006

### rakeshbs

maxwells third equations states that the curl(electric field)=-rate of change of magnetic field intensity... does it have any other physical significance other than being derived from faradays laws of electromagnetic induction... does curl(electric field) indicate the curling or rotational effect of the field??

2. Dec 20, 2006

### chroot

Staff Emeritus
Are you asking if the curl of the electric field is related to the curl of the magnetic field?

- Warren

3. Dec 20, 2006

### rakeshbs

i mean does it mean that a rotating electric field is produced due to a change in magnetic field... and a rotating magnetic firld is produced due to a change in electric field( from fourth equation)..

4. Dec 20, 2006

### chroot

Staff Emeritus
Yes to both.

- Warren

5. Dec 21, 2006

### daniel_i_l

that is how light propagates - when the E gets weaker the B gets stronger and when the B gets weaker the E gets stronger so the light can sustain itself without any external energy.

6. Dec 21, 2006

### Meir Achuz

"Rotating" is not the best description. The curl of a vector field is its rate of change perpendicular to its direction.

7. Dec 21, 2006

### rakeshbs

"Rotating" is not the best description. The curl of a vector field is its rate of change perpendicular to its direction.

cud u please explain this clearly?

8. Dec 21, 2006

### rakeshbs

my interpretation when i saw the fourth equation was that a circulating magnetic field is produced due to a current flow or a change in magnetic field... like a current carrying wire has a circulating magnetic field around it...

and a circulating electric field is produced due to change in magnetic field from the third equation..
or does the third equation merely state that the work done in a closed loop in a varying magnetic field is not zero since there is an emf induced in the loop..

these two equations are a bit confusing to interpret correctly.. :)
first two equations are ok...

Last edited: Dec 21, 2006
9. Dec 21, 2006

### Meir Achuz

MaxIII is the differential form of Faraday's law, just as MaxI is the differential form of Gauss's law.
A vector field can have a curl without being "circulating". For instance:

-->
--->
---->
----->
------>

has a curl, but the lines of force are straight.

10. Dec 21, 2006

### quasar987

The curl of a vector field $\vec{F}(x,y,z)$ can be writen as

$$\mbox{curl}\vec{F} =\left(\frac{\partial F_z}{\partial y}-\frac{\partial F_y}{\partial z},\frac{\partial F_x}{\partial z}-\frac{\partial F_z}{\partial x}, \frac{\partial F_y}{\partial x}-\frac{\partial F_x}{\partial y} \right)$$

But also in the more illuminating way:

$$\mbox{curl}\vec{F}=\lim_{R\rightarrow 0}\frac{\oint_{C_R}\vec{F}\cdot d\vec{r}}{2\pi R}$$

where $C_R$ is a circle of radius R. That is to say, the curl of F at (x,y,z) is the path integral of F around a tiny circle centered on (x,y,z) [divided by the lenght of its its circumference].

With this definition of curl, you can see intuitively why the vector field in Meir Achuz's post is not curlless, and more generally, why the curl will have maximum value at a given point when the vector field is circulating around that point, but that it is not necessarily zero otherwise. It is only in this broad sense that we mean that the curl is a measure of the circulation of a vector field.

Last edited: Dec 21, 2006
11. Dec 21, 2006

### rakeshbs

so does this mean that these 2 equations are just a way to say faradays law and ampere circuital theorem in the differential way... so these equations does not have a physical significance ?

12. Dec 21, 2006

### quasar987

We have that

$$\mbox{curl}\vec{E}=\lim_{R\rightarrow 0}\frac{\oint_{C_R}\vec{E}\cdot d\vec{r}}{2\pi R}$$

$$\mbox{curl}\vec{E}=-\frac{\partial \vec{B}}{\partial t}$$.

So the physical significance of the equation is that when the magnetic field at a point is not changing with time, then the electric field at that point is such that the above line integral is zero. Conversely, when B is varying, E is such that the line integral in the definition of the curl is non-vanishing.

13. Dec 22, 2006

### rakeshbs

ok... thanx.. i had this crazy idea tat the 3rd equation states tat the changing magnetic field acts as a source for the curling of electric field...

14. Dec 22, 2006

### rakeshbs

maybe this question is stupid.... :)
is there any physical significance in taking curl(curl(E)) when deriving the electromagnetic wave equation from maxwell's equations..

15. Dec 23, 2006

### Amr Morsi

Personal opinion after considerable scientific research

Dear buddy,

With respect to E and B, involved in Lorentz force, (or, D and H):
The real significance of variation with time as in Maxwell's Equations, is that whenever B is changing with respect time, E must be unzeroed vector and whenever "epsilon*E" (which is D) is changing with respect to time, "B/mu" (which is H) must be unzeroed vector. (There are cases, where there is magnetic field and E> = 0>.)

With respect to mathematical meaning of the curl:
It is a measure of the alignment of the curled field vector to rotational-pattern around this 3-dimensional point, with direction normal to the surface of rotation. Away from the meaning of the curl, Maxwell's Equations can be understood by the way of spreading of electromagnetic variations (waves) through space.

Yours,
Amr Morsi.

16. Dec 23, 2006

### rakeshbs

but how did maxwell come to a conclusion abt electromagnetic waves from these four equation?

17. Dec 24, 2006

### tim_lou

if you look at maxwell's equation in differential form:
$$\nabla\times\vec{E}=-\partial_t{\vec{B}}$$
$$\nabla\times\vec{B}=\mu_0 \vec{J}+\mu_0\epsilon_0\partial_t{\vec{E}}$$

in vacuum, J is zero (there is no current density).

Hence,
$$\nabla\times\nabla\times\vec{E}=-\partial_t{\nabla\times\vec{B}}=-\mu_0\epsilon_0\partial_t^2\vec{E}$$
then from there, you get:
$$\nabla^2\vec{E}=\mu_0\epsilon_0\frac{\partial^2 \vec{E}}{{\partial t}^2}$$

which is the wave equation. and the speed of the propagation must be:
$$\frac{1}{c^2}=\mu_0\epsilon_0\implies c=\frac{1}{\sqrt{\mu_0\epsilon_0}}$$

Last edited: Dec 24, 2006
18. Dec 25, 2006

### Amr Morsi

Yes Tim,

That's right. This is the whole story. Adding that: excitation to the medium is made in these wave equations by charges and currents (even if they are delta functions).

Rakeshbs,

Exactly as Tim illustrated. This wave equation (after adding excitations) can describe any electromagnetic field (away from exactness of Maxwell's Equations, QM, ......etc.). By the way, "div(D)=raw" may be needed further more when accounting for sources. Moreover, Magnetic Field may be deduced, mathematically, from E with one of Maxwell's Eqs. Macroscopic epsilon and mu are averaging material's effect on electromagnetic fields.

Amr Morsi.

19. Dec 25, 2006

### rakeshbs

ok... thanx guys..