# How to read Maxwell's Equations?

I was talking with a colleague about Maxwell's Equation, in particular Faraday's Law of Induction, and I realised that I did not understand the following. Usually, this equation is expressed as:

The induced electromotive force in any closed circuit is equal to the negative of the time rate of change of the magnetic flux enclosed by the circuit. - Wikipedia

In other words, a magnetic field that varies in time generates a circular electric field. Now I tried to express the same idea but backwards, that is, a circular electric field generates a magnetic field that varies in time proportional to the curl of the electric field. My colleague did not agree with this idea and said that the only interpretation is the one equivalent with the statement from Wikipedia.

This got me thinking and the reasons to back up my point of view are:

1. Since the relation in discussion is $\nabla \times \vec E=-\partial_t \vec B$ one should be able to read it both ways, since the $"="$ sign follows the properties of an equivalence relation
2. If I consider the relations $\vec D=\epsilon_0 \vec E$ and $\vec B=\mu_0 \vec H$, we have the induced fields $\vec D$ and $\vec B$ as functions of the field $\vec E$ and $\vec H$. The way I understand the term "to induce", it would much rather make sense to say that Field A generates The induced Field B, than The induced field B generates Field A. Going back to Faraday law, this would mean that the electric field generated the induced magnetic field.

Although the second point is based more on my understanding of the terminology, which might be subjective, I do not find the first one to have the same issue.

Which is the correct way of looking at these equations (same problem arises on Ampere's law too) or is there a book/written material that addresses this topic, that I can find?

Edit: So far the question received two answers that are based on the causality of the problem, both of them suggesting that the electric and magnetic fields should be considered as a connected thing (thus the electromagnetic field). I did not search for material suggested by AlbertB, and will do in the following hours, but I have a follow up for both answers.

If there is no delay when it comes to "generating" one field from another, because they are intertwined, wouldn't that mean that an electromagnetic wave should propagate with an infinite velocity? This question neglects relativity, and I consider having a flawed image on how an EM wave propagates.

• Oct 20, 2017 at 13:52
• The behavior of EM radiation one can observe in radio waves. Oct 20, 2017 at 16:35
• Just adding to the existing answers: a magnetic field causes a change in an electric field, and an electric field causes a change in a magnetic field. The rates at which these changes occur are specified by the equations. Oct 20, 2017 at 18:55
• And as for treating two fields as "connected things," I find this book cover is oddly satisfying for describing how we can look at one thing (electromagnetism) from two different perspectives (electricity and magnetism) and see different things: images-na.ssl-images-amazon.com/images/I/… Oct 20, 2017 at 18:56
• I would say that at the root of this question is about cause and effect. The fact of the matter is that in nature you can never observe one event causing another. You can only observe events. Causality is an interpretation that humans give to their experience. This is illustrated by the fact that causality appears no where in the fundamental physical equations. If $A = B$ we tend to give it a causal interpretation, either $A$ causes $B$ or $B$ causes $A$. But either interpretation is equally consistent with experiment. Oct 20, 2017 at 19:15

To think of one field "generating" or inducing another is not correct. The electromagnetic field is one entity. A changing magnetic field co-exists with a curling electric field. There is no time-delay, at a given point in space, between the existence of a changing B-field and the existence of an E-field with a non-zero curl.

In response to the further question. I didn't mention causality, which only enters the picture when changing currents and charge densities are introduced. Maxwell's equations can be used to derive wave equations of the form $$\nabla^2 {\bf E} = \mu_0 \epsilon_0 \frac{\partial^2 {\bf E}}{\partial t^2},$$ $$\nabla^2 {\bf B} = \mu_0 \epsilon_0 \frac{\partial^2 {\bf B}}{\partial t^2},$$ which show that the E- and B-fields both propagate at exactly the same (finite) speed $(\mu_0 \epsilon_0)^{-1/2}=c$.

In your question, there is the underlying issue of causality. Does the changing magnetic field cause a curl in the electric field or does the curl in electric field cause the magnetic field to change? It can be useful to paint a mental picture where one thinks of one thing causing the other, but in reality these two things are completely intertwined and can't be separated.

To understand the philosophical point better, please take a look at the writings of Ernst Mach, where he writes (analogously) about Newtonian mechanics and the concept of force as being an agent that causes acceleration as unnecessary.

https://en.wikipedia.org/wiki/Causality_(physics)

• I googled Ernst Mach and have found this book. Is this what you were suggesting? Oct 20, 2017 at 21:06
• Yes, the book is on the internet archive, and you might want to look at it there before you decide whether or not to get a physical copy of the book, as it is quite dated. Mach's philosophical about the ideas of cause and effect is well summarized by the comment of UtilityMaximiser. The relavent section in Machs book are The Economy of Science (p481) and the criticism of Newton's laws (p242). Oct 21, 2017 at 0:44
• Here's the link: ia902703.us.archive.org/22/items/sciencemechanic01machgoog/… Oct 21, 2017 at 0:44

Regarding the edit, note that the answers say that E and B are intertwined at each point in space. How E and B at point X influence E and B at point Y as a function of time is another question entirely. Say you switch on a current in a wire in a space empty of electromagnetic fields. At the wire, you will get E and B fields, determined by Maxwell's equations, as you now have a current source (which is time-varying). These will create inhomogeneity in the E and B fields in space as there are fields at the source, but no field away from the source. This gives non-zero spatial and time derivatives of the components of the E and B field. You need to solve the field equations to understand at what speed and how this will propagate. The field one meter away from the wire does not instantaneously become non-zero as all spatial and time derivatives of the fields are initially zero and there are no sources at that location. You need to wait until there are variations in the fields just before the 1 meter distance, ie introduce time derivatives and spatial derivatives, to start "feeling" the influence of the source. This is where the finite speed of propagation of disturbances in the field comes from.

It appears that there are two phenomena intermixed, and that might be the source of the confusion. Both, Rob and Albert, are talking about the properties of electromagnetic waves. While you are talking about the properties of current and charges, independent of each other.

If you have a wire loop(s) being "crossed" by a magnetic field, the magnetic field will induce a current in the wire, which creates a voltage difference at its end points.
If you have a voltage source at the end point of a wire loop, the electric field will induce a magnetic field perpendicular to the loop.

From these examples, one can see that the processes (therefore the equations) are reversible.

• The question does not mention currents and charges at all. Where these are involved you can talk about something "causing" something else, since charges can be mechanically moved. A magnetic field does not induce a current. A changing magnetic field induces a current. Oct 25, 2017 at 8:12