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I read this from Div, Grad, Curl and All That:

The second reason for introducing the electrostatic field is more basic. It turns out that all classical electromagnetic theory can be codified in terms of four equations, called Maxwell's equations, which relate fields (electric and magnetic) to each other and to the charges and currents which produce them. Thus, electromagnetism is a field theory and the electric field ultimately plays a role and assumes an importance which far transcends its simple elementary definition as "force per unit charge".

The first reason was basically about how finding the field first, then finding the net force on a charge due to the net field simplifies calculations, but I don't understand the second reason. What is this fundamental importance of an electric field that the author is referring to? (PS, I am unfamiliar with Maxwell's equations and classical field theory)

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  • $\begingroup$ Oh thats a mistake, I just edited it. Thanks. $\endgroup$
    – bluesky
    Aug 20, 2023 at 2:16

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One of the ways to understand this fundamental importance of the fields (the electric and the magnetic field as well) is to consider what happens when the fields change. The Maxwell equations are \begin{align} \nabla \cdot \bf{E} &= \frac{\rho}{\epsilon_0} \\ \nabla \cdot \bf{B} &= 0 \\ \nabla \times \bf{E} &= - \frac{\partial \bf{B} }{\partial t} \\ \nabla \times \bf{B} &= \mu_0 \bf{J} + \mu_0 \epsilon_0 \frac{\partial \bf{E} }{\partial t}. \end{align} These equations are always valid for any electric and magnetic fields $\bf{E}$ and $\bf{B}$, at any moment in time, and uniquely determine their behavior. Assuming you are not familiar with this language, I will try to explain the very basics of what these symbols mean.

As said before, $\bf{E}$ is the electric field and $\bf{B}$ is the magnetic field. $\rho$ is charge density, think of it as a function that assigns a number to each point in space and time, whose value expresses how much electric charge is present at that point. If in a particular place $\rho=0$, there is no charge there. If it has some positive value, there is some amount of positive charge. If it is negative, there is negative charge. The absolute value of $\rho$ expresses the amount of charge. $\bf{J}$ is current density. It is basically the same thing, but for electric current: the value of $\bf{J}$ at each point expresses how much current is passing through that point, and in what direction. $\epsilon_0$ and $\mu_0$ are just constants which are there because of the system of units (SI).

The symbol $\nabla$ measures variation in space and $\frac{\partial}{\partial t}$ measures change over time. With this, we can give a simple cartoon interpretation of the equations: The first one indicates that electric charge causes an electric field changing throughout space (think of the electrostatic field of a point charge). The second one is similar, but where the electric charge was there is now a 0. This says that magnetic charges do not exist, or, equivalently, that the magnetic charge is always zero everywhere. The third equation says that a magnetic field changing over time produces an electric field varying throughout space. The last one says that both an electric current and a electric field changing over time lead to a magnetic field varying in space.

One could make this picture more precise by differentiating between $\nabla \cdot$ and $\nabla \times$; each describes a different kind of spatial variation, but that won't be necessary for the main point, which is that the fields continue to have very interesting dynamics even if there are no charges or current present (in the vacuum). If we set $\rho=\bf{J}=0$ in the equations we end up with \begin{align} \nabla \cdot \bf{E} &= 0 \\ \nabla \cdot \bf{B} &= 0 \\ \nabla \times \bf{E} &= - \frac{\partial \bf{B} }{\partial t} \\ \nabla \times \bf{B} &= \mu_0 \epsilon_0 \frac{\partial \bf{E} }{\partial t}. \end{align} Even in a vacuum, you can still have electric and magnetic fields evolving and interacting due to these last two equations (of course, for you to have the fields in the first place there needs to have been some charges or currents somewhere in order to produce them, but we can assume that this happened long ago and far away, so that these charges no longer affect the fields directly). This is what happens in an electromagnetic wave: you have electric and magnetic fields varying in both space and time, producing a cyclical pattern that propagates through space indefinitely and independently of any charges. This is what we call light, or electromagnetically radiation.

After learning the fields evolve dynamically by interacting with each other even when no charges are present, it becomes strange to go back to the simple idea that the purpose of a field is just to facilitate the computation of the forces felt by charges. While they are useful for that, the fields are true dynamical objects that interact with each other and lead to a wealth of different phenomena in the real world. Electrostatics, where the interpretation of the electric field as "force per unit charge" makes the most sense, is only a small portion of the total amount of electromagnetically phenomena. You can check by setting $ \bf{J} = \frac{\partial \bf{E}}{\partial t} = \frac{\partial \bf{B}}{\partial t} =0$ in the equations that if no charges or fields change over time, you get just that "electric charge causes an electric field changing throughout space", which is just electrostatics, where there are no waves and you can get away with simply treating the electric field as a way to calculate forces.

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The reason that a field description is far more useful than a simple force description is because of the fact that information does not locally propagate instantaneously in the physical Universe, as far as we know: in fact, there's an upper limit to the speed of those local propagations, which we know as the "speed of light", $c$.

Consider what happens if we have two charges separated by great distance, both initially at relative rest, and we start wiggling one of them. The second charge, since it can still feel - however small - a force from the first, should start wiggling too. But that's information that we started wiggling the first one, since an observer sitting there could use it to conclude that such a far away thing started happening that wasn't happening before. Hence, due to the delayed propagation, it has to not feel that force until some substantial period of time later than when we started wiggling the first charge.

If we simply used a force law, where the only relevant variables are the charge positions, we would have a lot of trouble describing this situation. We'd need to use, for finding the force on the other charge, the position of the first charge at a previous time, where it's been long enough that information about it has been able to reach the second charge, and when you start getting more than just two charges, this idea - which can still be used fruitfully in simple situations like this (it's called the "advanced and retarded potentials" or eponymously the "Liénard–Wiechert potentials") - but beyond that, rapidly gets very nasty.

A much neater description obtains if, instead of trying to track all the times when various things happen, we consider the idea of "information propagating" a bit more literally, and imagine that at each point in space there's a little datum which gives the information about in what direction and how intensely a charge will feel an electric force or what kind of magnetic deflection it will have. Then we can imagine that, when these data change, they coax changes in turn only in those immediately next to them(*), and those in turn, change those next to them in the same way, and so on, and so forth. Then, in our test scenario, what would happen is that when we start wiggling the first charge, it sets up such a marching-outward front (a wave front!) of changing force information, moving out at $c$, and eventually those changes reach the datum that is present where the distant charge is located, and that charge then - and only then - finally begins to feel a wiggling force.


(*) Used loosely: in a continuum there are no "immediate next door" points, but we can consider the transport process as the limit of smaller and smaller discrete steps - that's what differential equations are all about.

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What the author is getting at is that not only is calculating the forces out of the fields easier, but writing down equations relating the electromagnetic forces to their charge/ currents configurations is also harder than writing down equations relating fields and their charge configurations. This is ultimately because $$F_{\text{coulomb}}=k\frac{q_1 q_2}{r^2_{12}}$$ depends "quadratically" on the charges; it's only when you fix one of them (effectively downgrading one of them to the role of a test charge) that the force becomes linear (in the test charge). In that sense, the prescription of using fields instead of the forces is, in my opinion, a convenient way of writing electromagnetism as a linear theory.

Remark: When I say that the force depends "quadratically" on the charges, I mean only insofar as you treat them as two "relevant" variables to the problem, such that if $q_1$ and $q_2$ were the sums of charges, then the force would "expand" quadratically. Turning the force into some linear function is then just a change of perspective: you take one of the charges, say, $q_1$, and decide to not keep tract of the effect it has on the other charges. Then what you're left with is the quantity $$E:= k \frac{q_2}{r_{12}^2}$$ which is the prototype for the electric field, and is also linear in the (distribution) charges. Then you end up with two linear quantities, one of them linear in the distribution charges, and another in the test charges. This is what I mean by taking the force, which is a quadratic problem, and dividing it into two linear problems.

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