As we know, In a circuit, simple or complex, electric fields created by surface charges move electrons which creates current which creates magnetic field which can be coupled to other lines and induce voltage if it is changing by time, which also creates current and this current also creates magnetic field and voltage drop along where it is coupled so on and so forth. Voltage and electric field are related just like currents and magnetic fields are. Changing electric fields create magnetic fields and the opposite is also true. I mean what is the independent variable here? Everything is like being affected by one another continuously. Is it power or energy or fields? What variable or thing I must I hang on to or take as starting point for better analysis of circuits?
-
2$\begingroup$ You are correct. It's not "one or the other". The electromagnetic field is a single entity. If you want to understand how it behaves, then you have to look at all the field components at once. The main problem that the student discovers while learning electromagnetism is that we are trying a "gentle" introduction to it by explaining the static electric field and the static magnetic field first. This, however, is a mirage that does you disservice in the long run because your mind gets used to thinking about components of what is one single phenomenon as two different phenomena. $\endgroup$– FlatterMannCommented Jun 6, 2023 at 21:16
-
1$\begingroup$ In a circuit, the "independent" variable, as you call it, can be either a voltage source or a current source. These have the same, certainly similar, role as in heat conduction the prescribed temperatures at the end of a rod. These are prescribed time varying "boundary conditions" that prescribe the current in any branch that contains no inductor, or prescribe a voltage drop between any two points not connected by a capacitor. $\endgroup$– hyportnexCommented Jun 6, 2023 at 21:16
-
$\begingroup$ For Maxwell's equations, you can have an independently prescribed current density $\mathbf J_s$ as a volume source in $curl \mathbf H = \mathbf J + \mathbf J_s +\partial \mathbf D/\partial t$ and prescribed surface sources or, equivalently, prescribed surface values of $E$ or $H$, as is frequently done in diffraction problems. $\endgroup$– hyportnexCommented Jun 6, 2023 at 21:20
-
$\begingroup$ Possible duplicates: Do Maxwell's Equations overdetermine the electric and magnetic fields? and links therein. $\endgroup$– Qmechanic ♦Commented Jun 6, 2023 at 23:33
2 Answers
To frame this a slightly different way, imagine a roller coaster moving along a track. It moves in all three spatial dimensions in general. As it moves along $x$, for example, it must also move along $y$ and $z$ in a very specific way because of the constraint of the track. So does $x$ motion cause $y$ and $z$ motion, or does $y$ motion cause $x$ and $z$ motion? Or perhaps $z$ motion causes $x$ and $y$ motion?
The answer is, of course, that viewing any one as being the primitive cause of the others is arbitrary and not particularly meaningful. They are separate degrees of freedom whose rates of change are related to one another by way of a constraint being applied.
In the rollercoaster example, it is possible to reframe the problem to eliminate the constraint. If we consider not $x,y,$ and $z$ separately but rather the distance $s$ along the track, then the constrained problem with three variables reduces to an unconstrained problem with one variable. This is one of the great advantages of the Lagrangian approach which one learns in an intermediate mechanics course.
One could ask whether it is possible to do the same with electromagnetism, and the answer is ... sort of. By reframing Maxwell's equations by introducing the scalar and vector potentials $\phi$ and $\vec A$, the constraints relating the electric and magnetic fields are incorporated automatically, and we've dropped from six fields $(E_x,E_y,E_z,B_x,B_y,B_z)$ down to four $(\phi,A_x,A_y,A_z)$.
However, in doing this we find a new problem - we have eliminated the constraint, but we have introduced a redundancy. If $(\phi,\vec A)$ is a solution of Maxwell's equations for a given scenario, then $(\phi',\vec A')$ is also a solution for precisely the same scenario if $$\matrix{\phi' = \phi + \frac{\partial \chi}{\partial t} \\ \vec A' = \vec A - c\nabla \chi}$$ for any scalar function $\chi$. The solutions $(\phi',A')$ and $(\phi,A)$ are physically identical. In order to obtain a unique solution, we need to write down some criterion for choosing the one we want; this is called a choice of gauge.
So the punchline is that in electromagnetism, you may either work with the (constrained) fields or the (gauge-redundant) potentials. Unlike the mechanical case, there is no middle ground between the two.
You noticed a problem with conventional wisdom: the statement that a changing E (B) field creates a B (E) field. This is a fine way to think when designing motors and inductors, or preparing for an EMP attack, it’s not strictly true (it’s not even causal).
The independent variables are charges, currents, and their time derivatives, on the past light cone. They create electric and magnetic fields in such a way that the time derivative of one is proportional to the curl of the other. See: Jefimenko’s equation.
Whether you consider the fields fundamental or the potentials, is a whole nother debate
