EM equations - am I missing something?

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Discussion Overview

The discussion revolves around the structure and interpretation of Maxwell's equations in electromagnetism, particularly focusing on the relationship between the number of equations and variables involved. Participants explore the implications of boundary conditions and the nature of the equations as differential rather than algebraic.

Discussion Character

  • Debate/contested
  • Technical explanation

Main Points Raised

  • One participant notes a perceived mismatch in Maxwell's equations, suggesting there are more equations than variables when considering the components of electric and magnetic fields alongside spatial and temporal variables.
  • Another participant argues that the system cannot be completely determined without boundary conditions, implying that the equations are necessary for defining the fields in electromagnetism.
  • A different viewpoint suggests that there are actually "-2" degrees of freedom, questioning whether some equations imply others and expressing confusion about the constraints involved.
  • One participant clarifies that the components of the fields are functions of position and time, asserting that these are independent variables rather than parameters, and emphasizes that the nature of differential equations does not require a one-to-one correspondence with unknowns.

Areas of Agreement / Disagreement

Participants express differing views on the counting of equations and variables, with no consensus reached on the implications of boundary conditions or the nature of the degrees of freedom in the system.

Contextual Notes

There are unresolved assumptions regarding the definitions of variables and parameters, as well as the implications of the equations being differential rather than algebraic.

Maxicl14
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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.
 
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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.
 
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?
 
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.
 
nasu said:
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.
 

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