Electromagnetic waves have a physical crest and trough as observed in microwaves and radio waves. I understand that is electromagnetic field vectors that wave, not the photon. But how do they wave? What does causes them to change direction?
[Physics] How do electromagnetic waves wave
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Related Solutions
Many of the things you write sound OK. But I wouldn't say that the other directions in the graph are mere strengths, they indicate the actual value (strength/magnitude and direction as well) of the electric and magnetic fields.
First note that technically the electric field is a vector and it is a field so it should have a vector (possibly zero at every point). Each vector has a head and a tail. You can think of the location of the tail as telling you the place where the field has a particular value. Then you can think if the difference between the head and the tail as telling you both the magnitude and the direction of the electric field at that point, in some specified unit system. So imagine a bunch of arrows all the same color the location of each is telling you where it is telling you the field and how the arrow points from there tells you the value.
Then draw the magnetic fields in a different color. And in both cases you can't draw a vector at every single point because it would just be too much to see them all.
For a plane wave traveling through vacuum there is a great deal of regularity. If the wave travels in the x direction then the electric and magnetic fields all point in the y-z plane. And their values only depend on time and on x, so you can draw just one longitudinal line and it will tell you about all of the values at all the locations. And since the electric fields and magnetic fields all point in the y-z plane you can reimagine the y-z plane as like the independent axis of a graph.
And for a classical plane wave travelling through a vacuum your picture should have the electric and magnetic fields both be strong together at one plane of a fixed value in the longitudinal direction and then at a different plane corresponding to a different fixed value of the longitudinal direction. Strong together and weak together and then strong again but pointing in the opposite direction.
If so, you get a picture much like the first one you drew (though that is just for one the electric and magnetic fields, a wave has both). That is all fine for a classical wave travelling through a vacuum.
For your other questions I think you should look at existing questions about the different topics.
From a quantum perspective it is much more complicated. From a quantum view there isn't an electric or a magnetic field, there is a photon field. When you have a large number of photons all in phase with each other it can look or act like an electromagnetic wave, but it is still different, and if you have a small number or they aren't in phase then it just truly different.
The electric and magnetic fields classically are stand-ins for saying how charges interact (though they do have their own energy and momentum, pressure and stress, etc.) And in quantum mechanics different interactions are possible and so the photon field is a stand in for saying how those different interactions go.
Since in quantum mechanics the charges don't have a location and momentum saying that their momentum changes based on the field where the charge is located just isn't going to be possible because you have none of those things. And if you can't verify the field's values at any location it becomes difficult to say it is there in that way.
We make the objects we need to predict results, for quantum mechanics we need different objects because we are predicting different interactions. Only in some limits (that only hold sometimes) do we expect quantum mechanical effects to start to approximately look like classical effects.
As for a transverse spacing in a Faraday cage stopping a wave that varies only in the longitudinal direction remember that you were drawing only the simplest case, the case of a plane wave.
In a plane wave the entire plane parallel to the y-z plane has the exact same electromagnetic field. This very simple (to describe) solution allows the wave to travel entirely in the x direction, over time each part of the wave simply slides over, the new value at distance $x=x_0+\Delta x$ over is the old value at $x_0$ a time $\Delta x/c$ earlier.
But no real wave is ever that perfect. They might expand in a spherical front instead of a plane front. They might have a central region where it is strong and it gets weaker farther out (like a Gaussian beam) some people even make beams where the energy and momentum travels along a cone so there is a central strong region but if you place a small object there the region beyond it still gets light because the light from farther out came from a base of the cone that started out farther from the central region.
Sort of like if you had some racers all lined up for a race but instead if running in their own tracks they ran towards the same point that is in front on the center runner. The center runner gets there first, but if parts of the center of the track are damaged the runners that left later will eventually close in and it won't matter that that the runners originally closer to the center got affected.
So a real wave has a more complicated wave front. And in fact if you are trying to avoid the wires of the Faraday cage you need your beam to be focused to not extend out too far in the direction transverse to the direction of propagation. And this isn't restricting the amplitude, this is saying that the actual wave shouldn't have field values spread out in the yz plane, it should be focused to a small portion of the yz plane.
Imagine you shot narrow beams all towards the same hole in the Faraday cage, you need very precision aiming each point has to has "its x" aimed very precisely to get in that same hole.
In general if you aim your initial propagation directions with accuracy on the order of the wavelength then you gave changed the wave.
Thus is because the direction of propagation isn't a magic thing you can freely associate with a point. It depends on how the electromagnetic field varied in space. When you field didn't vary in the y and z directions then the field travelled in the x direction. Now you are trying to get each region to aim slightly differently so you try to adjust the electric and magnetic field here and there to have each part aim just right.
Waves tend to spread, so if you aim some at that one whole, lots will be lost in other directions and what aimed well for one whole is nor going to aim for the other holes, so the Faraday cage succeeds at blocking almost all fields coming at it assuming the cage is big enough to have many many holes.
I think many of these question can be (and have been) asked as separate questions.
Fields
First you need to understand what a field is. There is a very good answer by dmckee on what a field really is which you can (and should read), but I'll try my own version. Mathematically, a field is something that has a value at every point of space and time. A typical example is temperature. The air in your room has a different temperature at every point and this temperature may change with time, so to each point in space and time we associate a number $T$. We might write $T(x, y, z, t)$, indicating that the temperature is a function of $x, y, z$ (space) and $t$ (time).
Temperature is a scalar field because at each point it is a scalar (i.e., a number). But we can have different kinds of fields. For example, the air in your room might be moving around, and so at each point it will have some velocity $\mathbf{v}(x,y,z,t)$. This velocity is a vector field, because at each point it has a magnitude and a direction (if you don't know what a vector is, picture it as a small arrow; the direction tells you which way the air is moving at that particular point, and the length of the arrow tells you how fast it is moving).
Waves
Air can carry waves, which we call sound. Sound is nothing more than a bunch of air molecules oscillating together in such a way that they carry energy from one place to another, in the same way that we see waves in water. With our fancy fields we can describe a wave by saying that at any given point the velocity is oscillating back and forth, and the phase of this oscillation changes as we move from place to place.
Temperature and velocity are fields that, in a sense, don't physically exist by themselves: they describe some property of a fluid, but it is the fluid that has physical reality, not its properties. But there are fields that are not a property of anything else, and the electromagnetic field is the most important among them.
Electromagnetic field
The electromagnetic field is described by two vector fields $\mathbf{E}$ and $\mathbf{B}$, called the electric and magnetic field respectively. For the purposes of light we can forget about $\mathbf{B}$ and just talk about the electric field. Just like the velocity of a fluid, this field can be represented by an arrow at every point in spacetime. Its physical intepretation is that if you place a charge somewhere, there is a force felt by the charge that points in the direction of $\mathbf{E}$ and is proportional to its magnitude. (Also there are magnetic effects but we're ignoring those). This is simply a more sophisticated view of the idea that like charges repel and opposite charges attract; instead of thinking of a force between the charges, we say that one charge creates an electric field near it, which is in turn felt by the other charge.
An electromagnetic wave is simply an oscillation of the electric and magnetic fields. At each point, the field's magnitude is increasing and decreasing with time. Wikpiedia has some nice gifs showing this process in time and space. The wavelength is a physical distance: it's the distance between two maxima or two minima of the field. The amplitude is not a distance, however: it measures how strong the field is, and so it is measured in units of field (Newton per Coulomb or Volt per meter for the electric field in SI units).
You can see in the usual pictures that an EM wave is a transverse wave; that is, the direction of the fields is perpendicular to the direction of propagation of the light. This is in contrast to a sound wave, which is longitudinal: that is, the molecules oscillate back and forth, and the move in the same line that the wave travels.
So, let's answer your questions:
a) The peaks and troughs are the points where the magnitude of the field is maximum in one direction or the other. As such, it doesn't make much sense to distinguish between peaks and troughs, because if you look from the other side they switch places.
b,c,d) A wave doesn't really take up space. There might be fields over a region of space, but the arrows you see in the animations don't have a physical length. They represent the magnitude of the fields, but they don't occupy physical space. Remember that there are two arrows (because of $\mathbf{E}$ and $\mathbf{B}$) at every point in space. As I've said before and has been said in the comments, wavelengths are lengths because they are the distance between two maxima, but amplitudes are not lengths.
The mental picture you describe in your question is, if you forgive me, a mess. You're mixing this description of EM waves with the quantum mechanical point of view, which is almost sure to lead to errors. QM usually deals in terms of particles, so the basic idea is that now light is thought of as a bunch of particles (photons), with a certain probability at each point in space to find a photon. The thing with quantum mechanics is that it's extremely weird and even the very best physicists have trouble forming an intuitive mental image of how it works. So please just forget about photons until you really understand the classical waves I've described in this post.
Best Answer
It is important to remember that an electromagnetic wave is exactly what it says on the tin: an electro-magnetic wave, i.e. it contains both an oscillating electric and an oscillating magnetic field.
When the amplitude of one of the fields decreases, it causes also a change in the other field, and vice versa. This is due to Maxwell's equations, which link the changes of electric and magnetic fields in time and space.
A helpful picture to visualize what is going on in such a wave is the following:
As you can see, the electric field alone propagates as a sine wave, as does the magnetic field. However, it matters that they are both propagating along the same direction and have the same phase, while their polarization (the direction their field vectors point in) are 90° offset.
Maxwell's equations specifically state that a change in space of the electric field ($\nabla \times \vec E$) causes a change in time in the magnetic field ($- \frac{\partial \vec B}{\partial t}$). The second equation goes the other way around: a change in space in the magnetic field ($\nabla \times \vec B$) causes a change in time in the electric field ($\frac{1}{c^2} \frac{\partial \vec E}{\partial t}$).