# Question about maxwell's equations

## Main Question or Discussion Point

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??

## Answers and Replies

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

- Warren

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)..

chroot
Staff Emeritus
Gold Member
Yes to both.

- Warren

daniel_i_l
Gold Member
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.

Meir Achuz
Homework Helper
Gold Member
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)..
"Rotating" is not the best description. The curl of a vector field is its rate of change perpendicular to its direction.

"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?

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...

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Meir Achuz
Homework Helper
Gold Member
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...
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.

quasar987
Homework Helper
Gold Member
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.

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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 ?

quasar987
Homework Helper
Gold Member
We have that

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

And Faraday's law says that

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

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.

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...

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..

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.

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

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:
$$\frac{1}{c^2}=\mu_0\epsilon_0\implies c=\frac{1}{\sqrt{\mu_0\epsilon_0}}$$

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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.

ok... thanx guys..