What is the physical definition of causality? Maxwell's equations give a physical relationship between the electric and magnetic fields $\vec E$, $\vec B$ at the same time, which some interpret as changes in one causes changes in the other etc. I find this confusing because to me, the cause of both is charge and cause should precede effect.
Therefore, how do physicists determine if there is a causal relationship between two physical quantities?
 A: As far as the electric and magnetic fields go, neither one is the cause/effect of the other changing, since the very presence of an electric or magnetic field depends on your frame of reference.  Rather there is a single quantity changing, the field strength tensor, $F_{\alpha\beta}$.  A change in one doesn't cause a change in the other, simply one thing changes.
Causality is a extremely gigantic subject, and people have different interpretations of what it means.  I think a fairly classical description is that events can only be causally connected, if the event under consideration as the cause is in the past light cone of the event that is considered the effect, and visa versa (except with the future light cone).  Then causation is determined primarily through inductive reasoning and correlation, though the arrow need not be $\iff$, rather I believe the necessary condition is correlation, that is $\text{causation}\implies\text{correlation}$.  More frankly, if you flip the light switch and a light comes on and those two events are in each others light cones, and you inductively verify after numerous different experiments that those two events are correlated, then you imply causation (note it's a deductive logical fallacy, but its completely legitimate as an inductive method, vis-'a-vi by the method explained above).  Though, you wouldn't do that with the light would you...
I hope this helps.
A: Physics About Physics = Metaphysics
This is a question for philosophy, since it deals in the very foundations of how physics is done.
The philosopher David Hume notes that it is impossible for us to truly know that causality exists:

Hence we may discover the reason why no philosopher, who is rational and modest, has ever pretended to assign the ultimate cause of any natural operation, or to show distinctly the action of that power, which produces any single effect in the universe. It is confessed, that the utmost effort of human reason is to reduce the principles, productive of natural phenomena, to a greater simplicity, and to resolve the many particular effects into a few general causes, by means of reasonings from analogy, experience, and observation. But as to the causes of these general causes, we should in vain attempt their discovery, nor shall we ever be able to satisfy ourselves, by any particular explication of them. These ultimate springs and principles are totally shut up from human curiosity and enquiry.
Elasticity, gravity, cohesion of parts, communication of motion by impulse: These are probably the ultimate causes and principles which we shall ever discover in nature, and we may esteem ourselves sufficiently happy, if, by accurate enquiry and reasoning, we can trace up the particular phenomena to, or near to, these general principles. The most perfect philosophy of the natural kind only staves off our ignorance a little longer, as perhaps the most perfect philosophy of the moral or metaphysical kind serves only to discover larger portions of it. Thus the observation of human blindness and weakness is the result of all philosophy, and meets us at every turn, in spite of our endeavours to elude or avoid it.

As Good as it Gets
We never know that billiard ball A striking billiard ball B "caused" B to move. We just expect it to happen because every time we test such a situation, that's how it works. Our belief in cause and effect is a prediction of the future based solely on inference from experience. This is, of course, a fragile kind of knowledge:

Let the course of things be allowed hitherto ever so regular; that alone, without some new argument or inference, proves not that, for the future, it will continue so. In vain do you pretend to have learned the nature of bodies from your past experience. Their secret nature, and consequently all their effects and influence, may change, without any change in their sensible qualities.

In the context of physics, you must simply be content with the simple answer that a magnetic/electric relationship has been found in the past. It's an explanation that works for now.
A: There is no need for physicists "determine if there is a causal relationship" because in this context the word "causal" is entirely subjective.  Your question addresses the distinction between proximate and ultimate causes - by way of analogy, if I (the charge) throw a ball (the electric field) at a lamp, was the cause of the lamp breaking (the change in the magnetic field) the ball or me?
That's a philosophical question, however; all that actually matters from a scientific point of view is whether a particular model: (a) works; and (b) is convenient.  Maxwell's equations certainly work, and (if you don't understand relativity) it happens to be convenient to suppose that changing electric fields create magnetic fields and vice versa.
It should perhaps also be noted that one of the assumptions inherit in your question is not strictly true: electromagnetic waves can be created without any charges being involved, e.g., Hawking radiation.
A: If we define some event which we will call $p$ to be the cause and $q$ to be its effect, then $p$ and $q$ should satisfy the following rules,


*

*$p$ implies $q$ but $q$ doesn't imply $p$.

*in the absence of $p$, $q$ shouldn't exist either.

*$p$ and $q$ shouldn't be simultaneous events as viewed from any inertial frame of reference. 
A: I may add to the above already-excellent answer, that the reasoning behind the introduction of magnetic field can be explained a little less mathematical. 
Basically, first thing to consider is an electric charge. When it does not move in our reference frame it does not produce any magnetic field, just electric one. Now, if it starts to move, we percieve its width (and width of spacetime) parallel to its moving direction to contract so that for every $dl$ we have $dl' = \sqrt{1-v^2} dl$ - Lorentz contraction. Now, the field lines of electric field also contract - and the closer field lines are - the stronger the field. This added influence is simply called 'magnetic field' but it is not a separate entity. So, magnetic field is a relativistic shadow of electric field. This is reflected in Maxwell equation, where charge density only apperas with the electric field, not the magnetic one. So, charge causes electric field and our relative movement w.r.t. charge causes our perception of this field to change. 
This, however, is only half of the truth. This is because changes in electric fields have been found to propagate in vacuum - these are electromagnetic waves. In vacuum, Maxwell equations become completely symmetric - and one can not distinguish cause and effect when it comes to electric and magnetic fields. This is where 'relativistic reasoning' has to take precedence over our classic intuition and we have to accept that those fields - however distinct they might seem - are just one entity - the electromagnetic field. 
A: Many of the answers given above are wrong. The word "causal" is not subjective. One of the answers includes the analogy:

if I (the charge) throw a ball (the electric field) at a lamp, was the cause of the lamp breaking (the change in the magnetic field) the ball or me?

But this is not a good analogy because the two cases are not at all similar. If a charge oscillates and produces a change in a magnetic field, there is a time reversed version of the process that can and does happen. This is not just a theoretical possibility, it is the sort of thing that people can in fact do in a lab. So there is a sense in which it is not clear which event is a cause and which is the effect. Causality between charges and fields can sometimes be made clear in the context of a specific explanation. For example, if you have a transmitter, you can control the charges in the transmitter and produce a field as the effect of controlling those charges. If you have an aerial you don't control the field around the aerial, the field around the aerial controls the signal produced by the aerial. But if all you have is a description of oscillating fields and charges in general you can't work out which is cause and which is effect.
There are several disanalogies between the ball and the fields. Throwing the ball at the lamp is not time reversible. Anyone who claims that it is time reversible is playing a word game that has nothing to do with the sorts of operation that can be done in reality. There is an objective sense in which the lamp would not break if the ball was not thrown. And there is an objective sense in which the lamp breaking is the effect and not the cause. And there is an objective sense in which the ball being thrown is the cause and not the effect.
Part of the problem people have with understanding cause and effect is that causality is a relatively high level abstract issue compared with equations of motion. Generally, cause and effect have to be judged in the context of an explanation of what has happened. You can't just crank the wheel on an equation of motion and have the right answer pop out.
Now, the electric and magnetic fields in a region can change without the charges in that region changing, so if you can't think in terms of electric and magnetic fields you're in trouble. Also, even in many cases where electric and magnetic fields interact with a material, and so interact with charges, it is possible and useful to treat what's going on without explicitly taking account of the charges. For example, reflection and refraction at the surface of a dielectric can be treated in terms of continuity of fields. There is an underlying explanation of refractive index etc. in terms of fields interacting with charges, but once you know that explanation you don't have to treat it explicitly in every case and doing so will get in the way of understanding.
One of the answers claims that we can't know that causality exists because we can't directly observe it according to Hume. The same answer also claims that we arrive at ideas by inference from experience, but this is false. (Another answer claims that we inductively verify ideas, which is another name for the same alleged process and is also false and extremely misleading.) For example, for a lot of observations of the motion of the solar system made before the 19th century, general relativity and Newtonian mechanics agree on their predictions. So there can't be a right way of inferring the right theory from observations since we have two contradictory explanations of the same events.
We know about the causal connection between two events in the same way we know about anything else. We guess about the solution to some problem, then we criticise the guesses until only one is left and it has no known problems. Our knowledge is not based on anything at all. It is guesswork controlled by criticism including results of experiments. See, for example, "Realism and the Aim of Science" by Karl Popper, Chapter I, which has a section refuting Hume. "Objective Knowledge" by Popper Chapter 1 refutes the idea that theories are inferred from experience. See also "The Fabric of Reality" by David Deutsch, Chapters 3 and 7. "The Beginning of Infinity" by Deutsch, Chapters 1,2,5 is also worth reading, especially chapter 5 part of which treats the issue of causality.
