# EM equations - am I missing something?

• I
• Maxicl14
In summary, there is a discrepancy between the number of equations and variables in Maxwell's equations, with 12 equations and 10 variables when parameter variables are set. This is unlike simpler equations, such as the simple harmonic equations, which have a 1 to 1 ratio between equations and variables. It is suggested that this could be due to the presence of additional degrees of freedom and the need to specify boundary conditions. The relationship between the equations and whether some imply others is also questioned.
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.

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.

## 1. What are EM equations?

EM equations, also known as Maxwell's equations, are a set of four equations that describe the behavior of electric and magnetic fields. They were developed by James Clerk Maxwell in the 19th century and are fundamental to the study of electromagnetism.

## 2. What do EM equations explain?

EM equations explain how electric and magnetic fields interact with each other and with charged particles. They also describe how these fields propagate through space and how they can be generated by electric charges and currents.

## 3. Why are EM equations important?

EM equations are important because they provide a mathematical framework for understanding and predicting the behavior of electromagnetic phenomena. They have been extensively tested and are considered one of the most successful theories in physics.

## 4. Are there any real-world applications of EM equations?

Yes, there are many real-world applications of EM equations. They are used in the design of electronic devices, such as computers and cell phones, as well as in the development of technologies like wireless communication and radar. They are also crucial in understanding natural phenomena, such as lightning and the Earth's magnetic field.

## 5. Are there any limitations to EM equations?

While EM equations are incredibly powerful and accurate, they do have limitations. They do not take into account quantum effects, so they cannot fully describe the behavior of subatomic particles. They also do not account for the effects of gravity, so they cannot be used to describe phenomena on a cosmic scale.

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