why do we need four maxwells equation?

E and H are two variables we want ot solve, but why do we need four equations to solve two variables E and H ....

Intriging issue. Stop thinking as a low level engineer.

Post the four Maxwell equaltion and put your notes of what do you understand from each one. Carefull: Not what you read from the book, what do YOU understand.

Then we will go a bit further...

D.

To make life a bit easier....a vector is completely specified if ad only if two parameters of the vector are known....find that and mathematically you are there.Intuitively lets resort to dkace way.....
A vector is just not any other " variable"

Well no exactly, in reality you have 6 variables and two equations.
plus you have E/H=120pi and the boundaries conditions.....
Why six variables?.......think about it a little bit.......
Is this a tricky question? or it is a sincere one?......

Hi, All:

I think it is due to the differential.

For algebraic equations, the Nu (number of unknowns) should be equal to Ne (number of equations) for a unique solution. For a differential equation, it is not true because some information is lost in the differentiation process.

Just take the simplest wave equation:

d^2 (Phi)dx^2 + k^2 * Phi = 0

The phi is not unique and you need extra conditions in order to solve the equations.

Regards.

i agree with jian's explanation.
A vector field could only be determined provided the curl and divergence of this vector are determined simutaneously and E/H is never seperated, but twisted.
The "extra conditions" you mentioned play the similar role as "boundary condition" for uniquely determining the final value of the field.

E and H are functions of space(x,y,z) and time(t). so i think we need 4 equations. dont think E and H to be variables.

amrith04,
Maxwell's eqns are for obtaining the solutions for E and H only right...
not to find x,y,z & t.....correct me if i am wrong

According to Helmholtz theorem, a vector which is zero at infinity is completely defined only if both divergence and curl are specified. Thus for two vectors E and H, we need 4 equations.

amrith04,
If u say that there are 4 variables x,y,z,t then i think u are wrong.....we have Ex,Ey,Ez and Hx,Hy,Hz component....so it comes to 6 variables?
correct me if i am wrong.....

We have 4 independent variables x,y,z and t. Ex, Ey and Ez are depandent variables. So we need 4 equations to solve the system with 4 independent variables.

hi,

actually you do not need the 4 maxwell equations. You can find E & H with only two of them. Since Maxwell equations are actually dependent:)

here is the proof

curl E = -j w B .... 1
curl H = j w D + J ... 2

div(1) ->
div curl E = -j w div(B) = 0 => div(B) = 0 (fourth maxwell equation)

div(2) ->
div curl H = j w div(D) + div(J) = 0

and using the continuity relation (div(J) = -j w rho)
you can get
j w div(D) - j w rho = 0 => div(D) = rho (third maxwell equation)

P.S : I used the vector identity div curl A = 0 (the divergence of a curl of any vector is equal to zero)

best regards,
Adel

For historical interest...Maxwell actually published the first complete form of Maxwell's equations in 1865, same year the US civil war ended. He wrote them in the form of vectors, but vectors (and, later, a related concept, quaternians) were not yet a formal mathematical concept. Thus, he actually wrote out each element of each vector. What we would today call one vector equation, he wrote as three equations.

In addition, Maxwell did not view E and H (or D and B) as primary. Those were secondary, derived concepts. He viewed something Faraday originally called "electrotonic state" as primary. Maxwell called it "electromagnetic momentum" because the time deriviative of this quantity is equal to force, just like mechanical momentum.

Today, we call this quantity magnetic vector potential. Maxwell assigned the letter "A" to it, presumably because he viewed it as primary. He then assigned the letter "B" to magnetic field, presumably because he viewed it as a secondary, derived quantiy (obtained from A).

Now, add in electric charge, electric potential, and electric current as variables, and it turns out Maxwell had 20 equations with 20 variables. They are: E,H,D,B,J,A,q,phi (scalar electric potential). Ironically, his work was virtually ignored in 1865, and it would take 20 years before Lodge, Fitzgerald, Heaviside, and Hertz took the time to figure out that Maxwell was not only right, he was an absolute genius. Maxwell died in 1879, before any of this happened. The four "Maxwellians" above (but primarily Heaviside and Hertz independently) then put Maxwell's equations into their modern vector form, which Maxwell himself never saw. But to honor Maxwell, they called the result Maxwell's equations. That was really decent of those guys, they did not have to do that.

Heaviside also added electric vector potential, F, and magnetic current, M (which does not seem to exist in this universe, but is useful mathematically). Heaviside initiated the view that E and H (or B) are primary, and that is what is taught today. However, physicists are returning to the view that A is primary, for some good reasons.

This means, today, we have an additional 7 variables (F, M, and magnetic charge). Why are there 27 variables and 27 equations? I do not know. (x,y,z,t are not variables in this sense. Rather, they are the dimensions of the space in which we solve for the variables.) And it is possible that the ultimate answer can not even be known by us mere humans. Amazing universe, isn't it?

A bit tired from travel (what else is new?)...forgot one more variable, magnetic static potential, total 28 variables, 28 equations (when written without the benefit of vector notation).