I EM equations - am I missing something?

[Maxicl14]

Summary:: There seems to be a mismatch, in the "Maxwell's" equations, between the number of equations and number of variables.

I was trying to play around with the equations for Electromagnetism and noticed something unusual. When expanded, there are 8 equations, 6 unknown variables, and 4 parameter variables. When the parameter variables are set using 4 more equations, the result is 10 variables and 12 equations. This seems 2 equations too many.

The variables I am referring to:
Ex, Ey, Ez, Bx, By, Bz,
x, y, z, t

The equations I am referring to:
1 from divergence of E
1 from divergence of B
3 from curl of E
3 from curl of B
4 from setting parameters: x=X, y=Y, z=Z, t=T.To give an example, the simple harmonic equations may be:
(d2/dt2) X(t) = -X(t)
t = T
This results in a unique solution of X (at t=T). 2 equations. 2 variables. Unique solution.

So why do Maxwell's equations behave differently, or do they? May it be to do with initial conditions? May some equations imply others?

Thank you, Max.

 

[Vanadium 50]

You are missing a lot, I'm afraid. Let's start with the biggest one: if you did not have additional degrees of freedom the system would be completely determined without needing to specify the boundary conditions. Therefore the system wouldn't depend on the boundary conditions, so E&M would not be useful - there would be fields that were what they were and that would be that.

 

[Maxicl14]

I understand that boundary conditions can be set, for example through the J function. Or by introducing another equation.
But here surely there are "-2" degrees of freedom, not 2 degrees of freedom, so the system seems oversaturated with constraints...
Do some equations/relations imply others?

 

[nasu]

Your counting seems weird. There are 6 functions of position and time (the components of the fields). Position and time are not parameters but independent variables. And you have 8 differential equations.
If you have one unknown function and one differential equation you cannot completely determine the function. These are not algebraic equations to have a 1 to 1 ratio between equations and unknowns.

 

[Maxicl14]

Your counting seems weird. There are 6 functions of position and time (the components of the fields). Position and time are not parameters but independent variables. And you have 8 differential equations.
If you have one unknown function and one differential equation you cannot completely determine the function. These are not algebraic equations to have a 1 to 1 ratio between equations and unknowns.

Oh ok. So 1 to 1 not necessary. Thank you.