Theory of relativity

I went back to K.L. for Chinese New Year celebration a few months ago. Over dinner, I was explaining the theory of relativity to my friends and to my utter dismay, I did a terrible job of it. And in the process of doing so, I realised that I did not understand the theory as well as I had hoped. So here I am again, attempting to explain this theory which I failed to do so miserably months ago and I hope this time, I would do a better job.

Motion is relative
We need to first understand that motion is not absolute, but relative. What do we mean by that?

Let me begin by a simple question: If you are standing still, are you moving? This question may sound absurd and to many, the answer is a quick "no". But what if I repeat this question to someone who is observing you from the Sun? The Earth revolves around the Sun, so if you are standing still on Earth, you must be rotating together with Earth about the Sun. So an observer at the Sun will see you as "moving", while an observer on Earth will see you as "standing still".

Let's look at another example. If you are at the train station when a speeding train whizzes past you. To you, it looks rather obvious that the train is moving - moving FORWARD. But to the passenger on the train, it is you who is moving BACKWARD. For the same motion, two different observers see it differently and there is no one single reference frame which we could base all our observations upon and agree! Therefore motion cannot be absolute. It must be relative, i.e. whether an object is moving forward, backward or not moving at all is completely subjective to the observer or the equipments that are measuring it. A motion or measurement of speed has no meaning if we do not attach a point of reference (or observing point) to it!!

Let's say the train is moving at 50 km/h. If the passenger in the train throws an apple vertically into the air, the apple would have landed back onto his hand. But to the person at the train station, it would seem like both the apple and passenger had moved forward in the same speed and thus the apple landed onto the hand of the passenger again when it drops. Now imagine that the passenger now throws the apple in the direction of the moving train at a speed of 10km/h, what would the person at the train station see? He/She would have seen the apple travelling faster than the train! The apple has to be travelling faster than the train in order for the passenger in the train to see the apple going forward. The speed of the apple would SEEM to be travelling at 50+10=60 km/h. In other words, the velocity of the apple is the "sum of the velocity of the train and the velocity of the apple being thrown":

v_sum=v_train + v_apple

v_sum is the velocity of the apple as seen by the person at the train station
v_apple is the velocity of the apple as seen by the passenger on board the train
v_train is the velocity of the train as seen by the person at the train station

If, however, the passenger is now throwing the apple in the backward direction, i.e. in the opposite direction of the travelling train. Then, the person at the train station would see the speed of the apple as "the speed of the train minus the speed of the thrown apple", i.e.

v_sum=v_train - v_apple

Speed of light is constant in vacuum
From the above, it would seem that 'speed' can be added linearly. The above is true for Newtonian physics - physics before theory of relativity. But the observant reader would have noticed that I emphasized the word "seem" to suggest that Newtonian physics is not entirely true, especially when we approach the speed of light. Motion is still relative, but the assumption that "v_sum=v_train + v_apple" no longer holds true! Let us see why.

Center to the theory of relativity is the discovery that the speed of light is constant in vacuum. In the late 19th century, carefully conducted Michelson-Morley experiment had proved this so. But how can this be true if speed can be added linearly, like the apple thrown in the train?

Imagine now that instead of throwing apple, the passenger turns on his/her flashlight and point it in the direction of the moving train. If the observer/person at the train station carries equipment that could measure the speed of light, what would the measurement be? Based on what we have discussed earlier, we would have expect that the speed of light measured by the person at the station is the speed of train plus the speed of light. Just like,

v_sum=v_train + v_apple

with v_apple now representing the speed of light as seen by the passenger.

but if v_apple is the speed of light, then v_sum would have to be greater than the speed of light. But this is NEVER the case! One may argue that perhaps the speed of the train is so small compared to the speed of light that the equipment is not sensitive enough to pick this up. But very careful measurements have been made and every time the speed of light remains constantly the same, whether it is measured by the person at the train station or the passenger in the moving train. This is very peculiar indeed and the assumption that speed can be added linearly must surely be wrong!

The physicists who came upon this experimental result were dumbfounded and they needed to re-visit their fundamental understanding of what 'speed' really means and how measurements are made (did they measure the speed of light wrongly?).

How do we measure distance and time?
There are many ways to measure distance but the basic principles are the same. We measure distance, for example, by using many rods of the same length and lining them up together. If we find the 3 rods lined up exactly along the side of the table, we say that this table is 3-rod long. The common 'rod' that we used is called a meter, so we can claim that the table is 3-meter long. But what if the table is just a little short of the 3 rods? How do we measure the length then? We could, of course, then use shorter rods. That is why we have centimeters, milimeters and inches. They are just 'rods' with different lengths.

Measuring time is a bit more difficult, but again the basic principles are the same. Time can be defined as 'how long it takes for an event to happen'. Now if we can find an event that always take the same amount of time to happen, we can use that as our reference to measure the time for everything else - just like the rod for measuring the distance. The first clock, invented in China, was based on the event of a dripping water. Today, most mechanical clocks measure time based on the event of the ticking of a second.

Special Theory of Relativity: Time and space is relative
From the discussion so far, we know that when we approach the speed of light, things are not always as they seem. For one, speed can no longer be added linearly and that the speed of light is always constant, no matter how you measured it. What does this tell us about time and space?

If the speed of light is always constant, something else must be changing. Speed is defined as the distance travelled divided by the time taken. If speed remains constant, then the distance travelled or the time taken or both must have changed in such a way that the speed of light is always constant. For example, if v = a/b, then v remains constant if a and b increase or decrease by the same amount. It turns out, after some mathematical considerations (called Lorentz transformation), that both the time and distance must have changed. This suggests that not only motion is relative, but time and space must also be relative (note that space is just length in 3-dimension)!

Even without the mathematics, but just logic considerations, we would have came to a similar conclusion. Let's now go back to the example of measuring light speed on the train. The passenger on board the train, travelling at the speed of the train, shines his/her torch light in the direction of the moving train. The person at the station makes a measurement of the speed of the light. According to Newtonian physics, he/she should obtain a speed faster than light in vacuum, but he/she does not. To correct for a smaller value for speed (v = a/b), either the numerator have to be smaller or the denominator have to be bigger or both. The numerator turns out to be the distance (smaller) and the denominator is the time (bigger), and so our measurements for distance and time must have changed accordingly. It must have because space has contracted, and time has dilated (moved slower).

Hence, the theory of relativity predicts that as we approach the light of speed, space contracts and time dilates (move slower), i.e. the measured distance is shorter and the time taken for the measurement is longer (since 'distance' or 'length' in 3-dimension is space).

But the adept reader would have argued that if space contracts for an object that is travelling at the speed of light, would not the rods - which we make measurements with - contract as well. Thus, we would have counted the same amount of rods for the contracted distance. The fallacy of the argument lies in the fact that this reader had failed to see where the measurement is being made. The measurement of the object travelling at the speed of light, is made by the person standing 'still' at the train station, using the rods provided. These rods, are not travelling at the speed of light. They are WITH the person, standing 'still' and therefore do not experience space contraction. So the measurement was made by using rods at still, to measure the distance covered by an object travelling close to the speed of light, which is experiencing space contraction. And the same explanation goes for the time dilation.

We have so far covered the "special" case of theory of relativity. So what this theory say is that not only motion is relative, time and space are also relative. There is no way we can say for absolute, 1 second is 1 second and everyone else agrees. 1 second is different for everyone. 1m is different for everyone. But at speeds much less than light, these differences are OK. There is, however, a bit more to this theory ...

The Twin Paradox
We can claim that A is moving away from B at light speed. But since motion is relative, this is analogous to saying that B is moving away from A at light speed. Remember, the relativity of motion states that there is no way we could know whether A or B is really moving, it only depends on where your reference is.

Consider this: Ali and Wong are twins. One day, Ali sat on an alien spaceship and whizzed off to outer space at light speed. After a certain time Ali came back to Earth, also at light speed, and met his twin brother Wong again. When they compared their watches, they realised that Wong had aged faster than Ali.

Let's consider the problem in detail and see why this is such a tricky problem.


What does Wong see?
1. Ali travelled away at light speed.
2. Since Ali was moving at light speed in relation to Wong, Wong observed that Ali's time ran slower (dilated). (For argument's sake, let's just assume that for every 1 hours that Ali had travelled, Wong observed that it only took 2 hour.)
3. Ali stopped his space ship when Wong's clock recorded 10 hours had passed. Due to time dilation, Ali's clock recorded only 5 hours.
4. Ali now return to Earth and Wong recorded that it took another 10 hours, but again, Ali's clock recorded only 5 hours.
5. In total, Wong waited on Earth for 20 hours, but observed Ali's clock to have only ran for 10 hours.


What does Ali see?
1. Ali travelled away at light speed.
2. Since motion is relative, Ali observed that he was stationary but it was Wong who was travelling away at light speed.
3. Ali stopped his space ship when his clock recorded 5 hours had passed. Due to time dilation, Wong's clock only recorded 2.5 hours. (because to Ali, it was Wong who was travelling at light speed and thus experiencing time dilation)
4. Ali travels back to Earth and recorded that it took him another 5 hours, but again, Wong's clock only recorded 2.5 hours.
5. In total, Wong waited on Earth for 5 hours and Ali travelled for 10 hours.

Herein lies the paradox: Upon returning from his travel, Ali thought Wong was younger but Wong thought Ali was younger. How can they both see each other as being younger than themselves? One of them had to be wrong! Carefully conducted experiments suggest that Ali (who travelled away in light speed and returned) is the one who should be younger. Then, where did our understanding of theory of relativity gone wrong? Surely a theory must be able to explain the results of an experiment!?

The main issue here is that because motion is relative, i.e. we cannot distinguish who is moving or who is at stationary. Ali sees Wong travelling away at light speed, but Wong sees the same of Ali too, just like in the train example above. Therefore, it is perfectly logical (if motion is relative) to claim that either of them had experienced time dilation.

But since experimental results suggest otherwise, there must be something different between Ali and Wong. What is the difference? How can physics distinguish them? Can we distinguish who is the one who is really 'moving away'?

No. We cannot distinguish who is really 'moving away' but they could distinguish who is 'returning'. Now, the adept reader should realised that so far all our attempt to explain theory of relativity has been made on constant speed. No acceleration (change of speed) was involved. If the train moved at constant speed, then neither the passenger nor the people at the station would be able to tell who is moving. But what if the train suddenly brakes? The passenger on board the train would see the people at the station going backward but suddenly losing this speed. The people at the station would see the passenger going forward but suddenly losing this speed too. But the people at the station will not feel the 'force' of the brake, the passenger will!! The passenger will lunge forward as a result of the sudden brake and will have to hold on to something to prevent injury. The people on the station feel nothing at all!

So while we cannot distinguish the two based on speed alone, we can distinguish them based on a change in speed, i.e. acceleration. Acceleration - that was the difference, that was what broke the symmetry between Ali and Wong. As Ali turned his ship around and returned to Earth, he experienced significant acceleration. And it would seem from this case (since Wong aged faster) that the effect of acceleration was to 'distort' Ali's observation on Wong. During the turnaround, Ali's time stood still while Wong's time ran for another 15 hours such that when Ali returned to Earth he recorded 10 hours of travelled time while Wong recorded a total of 20 hours (5 hours for Ali's outgoing and incoming flight + 15 hours of the turnaround). The acceleration had obviously no effect on Wong.

Now, after this 'adjustment', both of them would have agreed that Ali aged slower.

(source: http://en.wikipedia.org/wiki/Twin_paradox)

The above example is a bit extreme, of course. No one could travel at light speed and turnaround at light speed IMMEDIATELY. If the turnaround was not that extreme (but still in the order of light speed) then Ali's time would not have stood still while Wong's time ran for another 15 hours. Instead, Ali's time would just run much slower compared to Wong's time e.g. 1 hour of Ali's time to 15 hours of Wong's time. Therefore, acceleration seems to give similar effect to time (and length) as travelling near the speed of light would. Acceleration would make time run slower and length contracts too. This is the basis for the general theory of relativity!


General Theory of Relativity: Gravity changes time and space

The main difference between the Special Theory of relativity and the General Theory of relativity is the inclusion of the effect of gravity for the latter.

While explaining the twin paradox, I explained that acceleration had a similar effect on time and space (remember that length contraction is equivalent to space contraction). In science, if there is ABSOLUTELY no way to distinguish between two things, then these two things are said to be same. Is there a way to distinguish between the effect of acceleration and a gravity field?

The answer is no. In the twin paradox, Ali made an instantaneous turnaround to return to Earth. During this turnaround, he experience acceleration. The effect of this acceleration is analogous to the effect of gravity. During this turnaround, Ali would be pushed back to his seat just like a gravity field is pulling him. Therefore, if acceleration could cause time dilation and space contraction, so can gravity. They are equivalent.

We can now rephrase what we know about gravity based on our understanding of the twin paradox. The effect of gravity is that time dilates and space contracts for observers close to the gravity field. For the twin paradox, we could have well said that during the turnaround, Ali was close to the gravity field and experience significant time dilation while Wong was sufficiently far enough that he experienced none.

Let's assume that time and space are the two axis of a big fish net. If we place a big object (like a planet) at the center of this fish net, this will create a huge gravity field. If gravity causes time dilation and length contraction, you could imagine that the net has a higher "density" near where the planet is located. The net will look as though it is bent around that region. That is why general theory of relativity says that space-time is warped (bent) by gravity field. Now, if we roll a ball across the net, you could imagine that the ball will initially move in a straight line but then curve inwards as it approaches the bent. This is how general theory of relativity explains the gravity pull!

(source: nrumiano)

(source: http://www.ndsion.com/Siren/archive22/schaag22.html)

And that's all to the theory of relativity! The rest are just maths and more maths. I hope this have helped you to understand the theory better. There is another excellent website on this here.

2.4 Potential Difference

Previous sections:
1. Introduction
2. Electrostatic
2.1 Coulomb's Law
2.2 Gauss's Law of electrostatic
2.3 Electric field in materials

2.4 Potential Difference

Recalling that for electrostatic, the electric field must satisfy the below two equations at all points of space and time.

∇.D=ρ

∇xE=0

In this section, we shall set out to prove the second equation, which reads "the curl of E is equals to zero". As mentioned in section 2.2, the symbol that looks like an inverted triangle, ∇, is called nabla and it is actually a vector (d/dx, d/dy, d/dz). We shall defer the mathematics to later and try first to understand the physical significance of ∇xE=0.

The past few sections described, in general, the behaviour of electrical charges under the influence of an electrical field. Up to this point, we would already know that a positive electrical charge moves along the 'force lines' of an electrical field. The electrical field 'points' from a positive charge, say in point A, towards a negative charge, in point B. This is equivalent in saying that there is a potential difference between point A and B, which moves the electrical charge. Although the word 'potential' is common among science and engineering students, many take this word for granted and do not really understand what it means. Therefore, before going any further, I will attempt to answer the following two fundamental physics questions regarding 'potential'. However, if you are already an adept physics student you can skip this part and go straight to the section on 'potential energy' below.

What is potential (from physics point of view)?

In physics, it is said that energy is conserved. No energy is created or destroyed. It is only changed from one form to another. When you throw a ball towards a wall, it has kinetic energy. When the ball hits the wall and bounces back towards you, portion of the kinetic energy is converted to sound (that is why you will hear a 'thud' when the ball hits the wall); part of the energy is converted to heat due to the friction of the ball with the wall and air (although this is small); and some of the energy is also converted to heat due to atomic collisions within the ball when it deforms as it hits the wall (amount of energy converted here depends on the material of the ball). The total energy of the ball before and after the collision with the wall is exactly the same.

But a peculiar case happens when we lift a heavy object against gravity. Obviously when we lift an object against gravity, we need energy. But if energy is not destroyed nor created, then where does the energy go to? It is not sound or heat or any kind of energy that we can clearly measure. But the energy is clearly not lost because if we now release that object, it picks up speed, gains kinetic energy and thus falls towards the floor. The energy is actually 'stored' within that object and we say that the object has gained potential energy, i.e. potential to do 'work' or release energy.

How does a potential 'do' work?

We can never know the absolute amount of potential energy that an object has, we can only know or measure the difference. For example, when we lift an object a certain distance above the ground, we can only know that this object has gained a certain amount of potential energy, proportional to the distance it has been lifted. But what is the absolute energy it has? How much potential energy does it have in the first place? Is it possible that even before we lift the object, it already has a lot of energy stored in it? We can never know. This is because we cannot measure or observe this 'stored' energy directly. Therefore what is important is the magnitude of the difference of the energy, not the absolute value of it. Because we are only concerned with the difference and not the absolute value, it presents us with a flexibility to define what potential energy is.

If we now reverse the process i.e. we release the object from a certain height and let it fall to the ground. This object will release the 'stored' potential energy, which is equivalent to the difference of potential energy between the height it was released and the ground. In other words, an object will attempt to do work to release the 'stored' energy when there is a potential difference. This difference can be a postive or a negative one, depending whether it is A - B or B - A. It is, however, of no real physical significance whether it is positive or negative. In practice, it is common that we define the value of 'potential' mathematically such that energy is released when the object move from a high potential to a low potential, i.e. when A - B > 0 then energy is released and work is done by the system; but when B - A > 0, work needs to be done on the system instead. (although students should take note that there are special cases where it is defined that energy is released when moving from low potential to high potential, this is only a change of definition and does not affect what we observe in the real world).

A way to imagine this convention is to to take the analogy of river flowing from the mountain tops to the sea. River will always flow (i.e doing work) from a higher place to a lower place (i.e. high potential to low potential). When the surface is flat, the water does not flow (i.e. no potential difference)

Potential Energy

If an electrical charge move from point A to point B under the influence of an electrical field, we could have as well said that this is due to a potential difference between A and B. This potential energy must have been stored when we moved, for example, a positive electrical charge against the electrical field. Just as when we lift a ball off the ground and release it, the ball would fall back to ground; so would the electrical charge if moved against the electrical field and then released, it will move back to its original position.

This stored potential energy due to moving a charge against electrical field can be calculated by using the simple equation:

Energy = Force x Distance.

However, if the force acts along a curved line, then the distance which the force acts may then not be trivial to calculate. We can still do this by assuming that a curved line is formed by a series of small arrows. The smaller the arrows, the closer it matches to the curve. In this way, we can calculate the energy that is stored in an electrical charge by tracing the force, F, along the small arrows. Energy is now equal to force x distance, i.e. F . dx where dx is the length of the very small arrow (Note: both F and dx are vectors. By taking the dot product of F and dx, we find the component of F in the direction of the small arrows.) When dx becomes infinitesimally small, this becomes an integration: ∫ F. dx

but F=qE, therefore

q ∫ E. dx = -Energy (to move from A to B). The negative sign is to indicate that work is done when moving an electrical charge against electrical field. As described above, this is just a convention such that we all agree energy is released when a charge moves from high potential to low potential points.

Energy/q = - ∫ E. dx
V = - ∫ E . dx

Therefore, potential difference is just the energy per charge. Now bringing the dx over the other side of the equation and considering only a small change in potential leads to:

- dV / dx = E

In 3-dimension, d /dx becomes (d/dx, d/dy, d/dz) which is equivalent to ∇. This suggests that E=-∇V, where ∇V is read as grad V.

Now, it can be shown (mathematically) that a charge does not gain any net energy by moving from A to B and then back to A again. E.g. the charge gains energy when moving from A to B and releases this same amount of energy when going back from B to A. We call this a conservative field, i.e. the net energy gain depends solely on the end points (initial and final position). You can take an electrical charge on the planet Earth, take it to the moon and the bring it back to Earth but at 1 cm away from the original position - the net energy that it gains or loses is still due to that 1 cm distance. The fact that you had brought the charge to the moon and back makes no difference at all!

So if we move an electrical charge from A to 'somewhere' and back to exactly A, no net energy is gained (or equivalently, no work is done). Mathematically, we write this as

∫ E. dl =0

where the integration is taken over a closed loop. A loop is like a line where the beginning point is the same as the ending point, therefore there is not net energy gained. From Stoke's theorem we obtain ∫ E. dl = ∫∇xE dl =0, i.e ∇xE = 0, which is what we set out to prove.

At first glance, this seems like an immensely stupid equation because we are calculating the energy to bring a charge from A to B and then back to A again, which is essentially zero. It sure sounds more useful if we are actually calculating the energy from point A to some other point in space. But the purpose of the equation is to help us in characterising electric field in static conditions, it is suppose to give us a set of 'criteria', if you will, to fulfill if the electric field is static and that is exactly what the equation does. The physical meaning of the equation is to say that in the case of electrostatic, the electrical field is conservative - that is the potential difference is independent of path (travelled by the charge) and only depends on the displacement (initial and final position) of the electrical charges. We will see later that this is not always the case. It is useful to note that the statement "Electrical field is conservative" also gives information on how to calculate the potential difference between two points.

So the equations ∇.D=ρ and ∇xE=0, together with the bounday conditions, which we will talk about in the next section, are all the information you need to know about electrostatic field.

Kirchoff's Voltage Law

As a final note, it is important to note that the above relates to Kirchoff's Voltage Law, which states that the voltage drop in a (closed) loop is zero. This is a mathematical consequence of ∫ E. dl = 0. If we have a battery, however, then the electrical field is no longer conservative, the situation is no longer considered as electrostatic and therefore this equation is no longer true in such situations.

2.3 Electric field in materials

Previous sections:
1. Introduction
2. Electrostatic
2.1 Coulomb's Law
2.2 Gauss's Law of electrostatic

2.3 Electric field in materials

So far we have considered the electric field in free space. Obviously, if we now consider the electric field in materials (insulators or conductors), things will be quite different.

In the previous section, we shown that ∇.E=ρ/ε. Let's call this ρ as free charge since it represents the electric charge that is 'free' to move around (note: only free charge contributes to electrical current). Let us now assume that when we apply an electric field to the material, the material will modify the electric field either by strengthening it or weakening it, depending what kind of material it is. Let us represent this effect by introducing an additional charge, called the bound charge, ρ_bound. (These charges are not free and are 'bound' to the material and they cannot freely move about to generate electrical currents.)

One of the ways to imagine bound charge is to consider applying an electric field to a perfect conductor. Assume that the material consists of many electrical dipoles, i.e. opposite charges separated at short distance. (Such dipoles may exist, for example, in molecules with ionic bonding. The electrical field will displace the positive and negative ions slightly to create a dipole.) Now, further assume that these dipoles are not free to move about. They are fixed or bounded to the material. They can only rotate about their axis. What the applied electric field would do to these dipoles is to align them along the electrical field lines with the head of one dipole lining up behind the tail of the other dipole (refer to the diagram below). The result of this 'bound charges' aligning is that the internal electric field cancel each other out (due to the positive and negative charge lying close to each other). This result is important, so remember it: Electrical field inside a perfect conductor is zero. What happens if it is not a perfect conductor? Then, the dipoles cannot align perfectly and the electric fields will attenuate/diminish but it will not be completely cancelled out. You may also ask, what allows us to make such assumption about the electrical dipoles? Nothing, except that experimental results seem to suggest that such assumption is reasonable. As with previous, science is always about suggesting a good assumption to explain the experimental results.

Therefore, in the presence of any material (insulator or conductor), we modify the equation to become ∇.E=(ρ+ρ_bound)/ε to include the bound charges in a material. Having different kinds of charges (bound and free) in the equation is very confusing, this is why in Maxwell's equation all charges always refer to free charges. In order to be consistent with this, we rearrange to 'get rid' of the bound charge in the equation, i.e.

∇.εE - ρ_bound = ρ ;

∇. (εE + P) = ρ ; where ∇.P= - ρ_bound

∇. D = ρ ; where D = εE + P

The polarisation vector P, is the electric dipole moment density. By considering that an electric field causes dipoles to re-arrange in materials, one can calculate that the actual effect an electric field has on a material is to generate a 'surface charge' and a 'volume charge' which is related to P. You can refer to many textbooks on how this is calculated, or you can take my approach, which is just to assume P is a value (like kilograms is for weight) to indicate how much the electric field is affected by the material.

Since the electric dipole is induced by E, we may suspect that P is related to E too, and this is indeed the case. However, the relation may not be a linear one. In most cases we can assume it is linear, i.e. P=kE. But we write P=εχE, where ε=permeability of free space (as usual) and chi, χ=electric susceptibility.

Then D = εE + P = εE + εχE = ε(1+χ)E = ε . ε_r . E ;
where ε_r = (1+χ) is the relative permeability

If ε_r is independent on position, i.e. the same throughout the material, then the material is said to be homogenous. If it is homogeneous, then in general, D = ε . ε_r . E is in matrix form where D and E is a 3 x 1 matrix and (ε . ε_r) is a 3 x 3 matrix. If only the diagonal elements of (ε . ε_r) is non-zero, i.e. the relative permeability is only dependent on the principal (x, y and z) axes, the material is called biaxial, or isotropic:
Dx = ε . ε_r11 . Ex
Dy = ε . ε_r22 . Ey
Dz = ε . ε_r33 . Ez
(where the number indicates the position in the 3 x 3 matrix)

Furthermore, if ε_r11=ε_r22, then the material is said to be uniaxial.

In summary, in the presence of (any) material, the electric field will be different than from the free space and this difference is accounted for by using ε_r, the relative permeability. The equation that relates relative permeability to the electric field, E and displacement field, D is

D = ε . ε_r . E.

But this equation can be confusing sometimes. We can essentially move the relative permeability to other side of the equation and it will now look like this: E = D / (ε . ε_r ). So does the relative permeability serve to modify D or E to account for the presence of a material? If, for example, a dielectric material is placed between two conductor plates (like a capacitor), a constant (electric or displacement?) field will be generated across the dielectric material. What happens at the interface between free space and this dielectric material? Does the displacement field or the electric field change due to the presence of this material? Or both? Although we can use sheer mathematics to find out the answer, it is much more meaningful if we instead rely on our intuition to understand why and which should be the answer. Obviously if both D and E change, and by the same amount, then there is no difference between the two quantity, so this is not allowed. The equation tells us that D and E is related by the permeability, but it does not tell us which is the constant and which is being 'affected' in this case. But if we pay attention to the words I have used so far, I have always said "the materials affect the electric field" and NOT the displacement field. This is a very reasonable statement, since inside a material, especially a conducting one, the charges are BOUNDED, NOT FREE. Displacement field's relation to the charge as indicated by Maxwell's equation ONLY refer to free charges. The free charges, accumulated at the surface of the conductor plates, are constant and therefore D should be constant. The electrical field, on the contrary, is related to the free charge AND the bound charge inside the material. And therefore it is the electric field that will be affected in the presence of a dielectric. (note: there are no free charges INSIDE a conductor but free charges can reside at the SURFACE of a conductor)

In short, the difference between D and E is that "D is the field due to the free charge only" and "E is the field due to both the free and bound charge", the effect of the bound charge is included indirectly through the relative permeability. The quantity D is introduced so that we can make Maxwell's equation look much neater, i.e. always only referring to free charge only. But the usefulness of this displacement field will be more obvious when we disscuss the dynamics of electromagnetism.

Although we have introduced many terms like the polarisation vector, the electric susceptibility, and the relative permeability, it is the relative permeability that is most commonly used to describe the effect of materials on electric field. However, we must always bear in mind that relative permeability is obtained through a series of assumptions. There will be time when we cannot use the relative permeability but instead must use the 'original' equation that contains the polarisation vector or susceptibility, especially for the case of describing in depth behaviour of materials. As a special case, consider a perfect conductor. What is the relative permeability of a perfect conductor? Because D = ε . ε_r . E and E is zero inside a perfect conductor, D will always be zero inside a perfect conductor but this is not true! If we instead use D = εE + P, then when E is zero inside a perfect conductor, D = P. The polarisation vector represents the contribution from the bound charge and thus D is non-zero even in a perfect conductor. Remember, relative permeability is useful because it is a GOOD APPROXIMATION to the 'overall' behaviour of electrostatic systems but it does not work all the time.

The end of knowledge

Of all the abilities that a man possesses, nothing is more important than the ability to transfer knowledge from a generation to the next. Human's life-span in the early days must have been somewhere between 20 -40 years old. Without the ability to transfer knowledge to our next generation, there would be very little progress in the development of our civilisation.

Through language, we transfer our knowledge in the form of story-telling. This is the earliest method that we use to transfer knowledge. But we all know that this is not a very efficient one because stories that are passed on from generation to generation are very susceptible to errors. After a few generations, the border between facts and fiction blurred, real knowledge became fairy tales.

Then, we developed writing and books. This was an important milestone. Information and knowledge that are contained within books or other form of earlier writings (whether it's on the rocks or tree skins) survived for many generations. Information was passed on accurately, withstood the harshest weather and test of time. Even now, we can still see discoveries of caves with ancient writings that are more than 3000 years old.

About a century ago, camera allowed us to store information and knowledge in the form of a picture. And more recently, the development of optical devices like CD, DVD and magnetic storage devices like MRAM, our thumbdrives and HDDs have revolutionise the way we keep information and thus our ability to transfer knowledge.

Our photos, 1000s of them, are in our camera, memory cards, flickr or thumbdrive and some in our portable HDDs; Our diaries, are on the internet - blogs, homepages, etc - which are stored in the server's HDDs somewhere in the world; Our contacts and addresses and to-do-list are on Excel spreadsheets, notepad, or some other form of electronic application stored in our thumbdrive which we carry with us all the time; Our books are in pdf format, stored in our computers; Our birth certificates and other important documents are scanned into tiff format and stored in HDDs or other magnetic storage devices; Libraries all over the world are turning into an electronic one, e.g. storing older books (that are already crumbling) in the formed of scanned digital copy.

I'm sorry I've written a rather long introduction, but yes, until here, it's only the introduction. Because the important point I want to make is this - are the methods of information/knowledge storage improving? Are CDs, DVDs and e-books better than the plain old paper books? The advantages of digital data storage are obvious. They retain information that doesn't fade with time, e.g. a typical paper book printed in 1990 would probably by now have a few pages that are fading and becoming brittle, but ebook doesn't suffer from this; Digital data are easily transferred from a point to another; They are easily managed because they are composed of a series of 1s and 0s; Their capacity are huge, e.g. a few DVDs the thickness of a regular book could probably fit in all the information contained in a typical school library. It would seem that our methods of storing information are actually better and have improved. So it seems that there are little signs that our capability to store information and knowledge would end.

But here's the catch. Two, in fact. And they go hand-in-hand. Firstly, although the writings on books and rocks may fade over the years, they are still legible after thousands of years. Digital data storage on the other hand stores information in, well, digital format which means that one error (a flipping of 0 to 1 or vice versa) would make the entire data corrupted and completely useless (FYI, I know there is error coding, but it isn't sufficient). Secondly, and more importantly, how long do you think a USB thumbdrive, HDD or CD could retain information?

USB thumbdrive? 1 year. Yes, after 1 year, it will lose some or all the information stored in it. Transistors-based storage devices like USB stores information by storing charge. When they are not in use, some charge will be leaking. Although small in quantity, but over time it could be significant enough to flip the data bit from 1 to 0. That is why, if you read carefully the instructions booklet, they always ask you plug it into a PC at least once a year. This is to restore/recharge the charge.

CD, DVD and other optical disks stores information by having 'dents' that reflects lights differently, thus indicating a 1 or 0. Over time, about 20-30 years, these dents loses their ability to reflect lights properly. Because these things are built with such 'accuracy' and each 'dent' is very small, any slight degradation in the material is sufficient to reflect the light wrongly. Therefore, again, flipping a 1 to 0.

Magnetic storage devices (like our HDD) stores information on magnetic particles that have tiny magnets pointing in either two directions, thus indicating 1 or 0. But over the years, heat will cause these tiny magnets to slowly rotate and point in a random direction. After about 100 years, the magnets will be sufficiently random that our magnet heads would not be able to determine if it was a 1 or 0. Our digital data is once again, lost.

In fact, I've just read a journal paper that did a study on the reliability and the life-span of our modern information storage devices. The outcome of the study is that most of our so-called advanced storage devices will not last up to a century. Therefore, it advices that these devices need to be periodically 'updated' or 're-backup'.

But what if there is a war. A terrible war. World war 3, perhaps, that was fortunately not to wipe-off the human race but did a terrible damage to countries all over the world. Would the backup and update happen? It has happened in our history, war-torn countries have their libraries burnt. But some surviving books and cave-writing or other forms of 'older' information storage devices was buried under the sands and rocks, waiting to be discovered by archaeologist many many years later. But is this possible with our USB thumbdrive, CD, DVD and HDD? I'm afraid not.

Our reliance on these so-called modern technology would one day spell the end of our knowledge. All will be lost. All the archaeologist could find are rubbles, pieces of broken CDs and thumbdrives, where all the information about our once glorious civilisation will all be gone.

Of course, this may not happen. If we are able to avert a worldwide massive war or if we invent a 'better' device. But I doubt the former will happen. And seeing the trend of current technology and engineering to prefer simple, easy devices with little long-sightedness, I doubt the latter will happen too. Think about it, we always design, develop and engineer products so that they are fancy, sellable, funky, fast, easy and lasts long enough until the next product is out - which is approximately 3 years or even shorter for some devices. My first PC lasted me 10 years. My last PC lasted only 2 years; The camera my dad bought when I was still a kid lasted 10 years, my last digital camera lasted 3 years before becoming obsolete.

Yes, the end of knowledge will be here. Pray that it will not happen in our generation. Or the next.

2.2 Gauss's Law of electrostatic

Previous sections:
1. Introduction
2. Electrostatic
2.1 Coulomb's Law

2.2 Gauss's Law of electrostatic

In the previous section, we have came to conclusion that E field is q/(4πεr^2). But that was due to a point charge and assuming that the field spreads out equally in all direction, i.e. in sphere-like manner. What if we want to know the field generated by an arbitrary collection (or shape) of charges and the field distribution other than a sphere?

This is where calculus comes in handy. We could always start with a small surface, dA and small charge, dQ and then sum it (integration) to obtain a more general equation for the electric field.

Recall that we obtain 4πr^2 from the surface of a sphere. Instead of using 4πr^2, let's use dA to denote an arbitrary small surface and ρdV to denote an arbitrary small volume of charge (where ρ is the charge density and dV is the small volume occupied by this charge). Then the E field is

E=(ρ * dV )/(dA * ε) ;

(quiz question: why don't we multiply a density function for dA like we did for dV?).

Rearranging and integrating,
ε∫E.dA = ∫ ρ dV

Here, it is important to introduce a very useful and important mathematical tool called Gauss' theorem. We need not concern ourselves how to derive this theorem. All we need to know is that this theorem converts a surface integral of a vector field into a volume integral, and vice versa.

∫F.dA = ∫ ∇. F dV

where F is any vector field and the sign ∇, called nabla or del, is the vector (d/dx, d/dy, d/dz). The term ∇.F is also called the divergence of the vector field F.

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At this stage, it may be important to ask what does divergence and the symbol nabla physically mean. The origins of nabla, divergence, and the curl of a field (not yet touched upon here) are used so often in field theory that I would dedicate a section to explain them. Please refer to this section for more information.
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Hence, applying Gauss' theorem on the LHS,
ε∫∇.E dV = ∫ ρ dV

which implies
ε∇.E=ρ or ∇.E=ρ/ε

Here, I will introduce yet another quantity called the displacement field, D. I will explain in more detail what it means in the next section. But for now, it's suffice just to remember that D=εE.

Hence, the above equation becomes
∇.D=ρ
which is the first equation of electrostatic.

Gauss theorem, is not a law of nature as some students may misunderstood. It is really just a fancy mathematical tool to convert a surface integral to a volume integral. Another way to see it, is that it converts the equation from a 2D one, to a 3D one and vice versa. For those who are interested you can look up for Green's theorem, which is a more general theorem to Gauss' and Stoke's.

It's always useful to take a step back and think how we've come to this equation to understand the physical significance of this equation. What ∇.E=ρ/ε really means is that the flux of electric field through a closed surface is equal to the total charges contained within the closed surface, multiplied by a constant. In a simpler form, EA=kQ, where E is the electric field, A is the area crossed by the electric field, Q is the amount of charge contained within A, and k is a constant. The product of E and A is also known as the electric flux.

The original meaning of flux is flow - as in flow of water. In science it usually means the rate of change of a particular 'thing' over an area. E.g. the flow of water in our pipes; flow of temperature; or in this case, the 'flow' of electric field. So, one can think of electric charges as sources where electric field will 'flow' from. And that from the principle of conservation of field (or matter), the source must equal to the resultant flow, i.e. total flux=source=kQ, which is the same as above (the constant k is just a scaling factor and can be easily set equal to 1 if appropriate units for Q and flux are used. Refer to previous section on how it was decided to use 1/ε as the constant). Just like our water supply, the amount of water that has flowed out from the pipe must equals to how much water is lost at the supply tank. All the fancy equations about fields come down this simple analogy of water flowing from a tap!

Different sides of magnetic and electric field

Did you know that electric fields and magnetic fields are really two sides of the same coin?

This is why...

A static electric charge does not generate magnetic field. On the contrary, an electric current, or moving electric charges, 'generates' magnetic field. But motion is relative to the observer.

So if we have a static electric charge at the center of the room and you stood still - no magnetic field is observed. But then when you started running in the forward direction, the charge would seemed like it is 'moving' backwards. And, as above, a moving charge 'creates' magnetic field! Therefore, by running around the room, your motion could actually 'create' magnetic field, despite the fact that it is just a static charge in the middle of the room.



But surely, from the example above, you should know that we did not really 'create' anything. We are merely observing the effects of moving/accelerating against a static electric field, i.e. magnetic field radiation is just the observation of disturbance to the static electric field. Hence, electric field and magnetic field must be just two sides of the same coin!

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There is a difference between observing static magnetic fields and observing radiating magnetic fields (as in the case of accelerating against a static charge). It is still an ongoing debate among physicist whether a uniformly accelerating static charge does indeed radiate. But if acceleration is not uniform, radiation is guaranteed. Those who are interested could search more information on the internet for this.

2.1 Coulomb's Law

Previous sections:
1. Introduction
2. Electrostatic


2.1 Coulomb's Law

Most scientific theories up to the late 19th century are based on Newtonian forces. So it's not unusual that the investigation of electrostatic should also begin with the force that is acting on an electric charge.

Let's begin our little experiment by isolating a single sphere of negative charge.

***********************************************
We like to assume these charges are sphere-like because we assume that they exert a force in all directions. But how does a charge actually looks like is altogether a very complex question to answer. Also, beware of the term 'single charge'. What is 'single'? Is there such thing as one charge? Without units, these terms become ambiguous. But for the moment, let's assume that there is such a thing called 'unit charge'.
***********************************************

Now, we all know from A-level physics that like charges repel and opposite charges attract. So if we then place a positive charge near to it, then both will experience a force that pulls them together. But for the moment, let's assume that the first charge, q1, is fixed in spaced and concentrate only on what happens to the second charge, q2.

To visualise this, we represent the direction of the force with an arrow. We repeat the experiment a number of times, each time with q2 placed in a slightly different position such that the end of the arrow coincides with the beginning of another arrow, this is what we might get.
Imagine if the experiments are repeated so many times and that the arrows become very small, it then forms a continuous straight line. If we place q2 in a space with two charges of opposite polarity, it becomes clearer as to what we are trying to achieve here.
These lines that we traced by repeating the experiment numerous time are called lines of action or more commonly known as field lines. They represent the direction which the force will be acting. In other words, if a charge is placed on a particular line, the force will act on it such that it will move along the line.

***********************************************
Generally, fields can be scalar or vector. In this case, electric field refers to the action of the force that pulls or pushes a charge. Force is a vector. Therefore, the electric field is also a vector.
***********************************************

Example of electric fields

So field lines are really just imaginary lines that scientists cooked up to explain what they observe in the experiments. They do not physically exist! The field lines are there just to help us to visualise. One can imagine that by placing a charge in space, field lines will emanate from it and exert forces on other nearby charges in the direction of the field lines.

Since electric fields are just some sort of forces, we can now choose to define it as the force per unit charge. We could have define it any other way (as long as it tells us something about the force), but scientists have found that this definition is the most useful (and they were right!) so we will just follow. But when we say force per unit charge, which charge are we referring to? q1 or q2? This depends on which charge the force is acting on, i.e. in this case it is q2. In other words, electric fields will tell you how much force a charge (q2) will be experiencing when it is placed in an electric field. We now have

E=F/q2

where E is the electric field (Newton per Coulomb), F is the force (Newtons) and q is the charge (Coulombs).

***********************************************
The reader may notice that we have not define the unit of Coulomb or how much is a single charge. We will come that later.
***********************************************

We now turn our attention back to the electric field. What causes it? The answer is obviously q1. Since the field lines are 'generated' by the presence of the charge, q1, it is natural to think that if q1 is 'stronger' or 'has more quantity of charges', then the field that emanates from it will be 'stronger' and the forces of attraction or repel will also be stronger. Just like in gravity, the bigger the mass the bigger the pull. And so it is reasonable to say that the electric field, E, is proportional to q1, i.e.

E = k . q1

Charles-Augustin de Coulomb, a French physicist, postulated that the force between two electric charges, just like gravity, has the inverse square law (i.e. proportional to 1/r^2) and experiments confirmed that his postulate was indeed correct. On this postulate, we can now further improve our above equation to

E = k . q1 / (4*π*r^2).

Hang on a minute! I said "inverse square law". I did not mention anything about the π. So where does the 4*π*r^2 comes from? Now, imagine a pulse of energy emanating from a single point. As time progresses, the energy spreads out in a growing sphere. The energy resides only on the surface of the sphere. Then, the energy density could be easily calculated as

Energy / Surface of the sphere = Energy / 4*π*r^2

Electric fields are not energy. But scientists found that other than energy, many of their observations in nature also obey this law. So, it was natural for Coulomb to assume electric fields behave like this as well. So, I emphasize again, that the inverse r^2 dependence is merely a postulate, or just a guess, if you will. This postulate was confirmed by experiments in Coulomb's days, and also with high-precision experiments of more recent days up to about 10^-12 cm and possibly even smaller. Unless an experiment can show otherwise, this postulate will hold.

A quick recap of what we have know so far,
F = q2 . E
E = k . q1 / (4*π*r^2)

which gives us,
F = k . q1 . q2 / (4*π*r^2)

Since at the point of this experiment in history, the value of k and unit charge (for q1 and q2) have yet to be decided, it is at our own discretion to choose a suitable value for them, as long as it matches with the observations in the experiments. But, as always, the scientists whom discovered the phenomenon have done this for us. For historical reasons, the value for k = 1/ε_0 and the unit for charge is Coulomb, as some of you may already know.

*************************************************
ε_0 is known as the permittivity of free space and the value is approximately 8.854x10^-12 and the unit is Farad per meter.

While 1 Coulomb is defined this way: If at each end of a meter is 1 Coulomb of charge, each charge will experience a force equivalent to 9x10^9N.
*************************************************

Plugging this in, we finally have the equation for the famous Coulomb's Law,

F = q1 . q2 / (4*π*ε*r^2)

and the corresponding electric field is

E = q1 / (4*π*ε*r^2)

This equation gives only the magnitude. Since electric field and force is a vector, we denote a_r as the unit vector pointing in the direction of where the force is acting. So, finally we get

F = a_r . q1 . q2 / (4*π*ε*r^2)
E = a_r . q1 / (4*π*ε*r^2)

It is now worthwhile to note that in our example above, it was assumed q2 did not 'generate' any field of its own when placed next to q1. In reality, this is not true. q1 and q2 both 'generate' their own fields and by placing q2 next to q1, their field lines superimpose to form new field lines. It is this new field lines that will determine the forces and how these charges pull or push each other. As you can imagine by now, things get messy very quickly when the number of charges that we consider becomes more than 2.

2. Electrostatic

Previous sections:
1. Introduction


2. Electrostatic

In this chapter, we will only concern ourselves with electrostatic, which means that there is no rate of change of electric charges with respect to time (In other words, as time passes by, the quantity of electric charges remain the same. Another way to look at this is to say that it is in equilibrium). This definition is important and the reader should always be careful that this criteria is met when applying theories and equations from this chapter.

It is also important know our objective of this chapter. What do we expect to know at the end of this chapter? We expect this:

∇.D=ρ and ∇xE=0

which are the two fundamental equations in electrostatic.

Looks simple enough! At the end of this chapter, I hope the reader would have understood how these equations are formulated and what do they mean physically.

We will begin by seeing how the investigation of force acting on electric charges (in vacuum space) leads to electric field and Coulomb's Law. Applying Gauss's Law, we shall see how does this gives rise to the first of the two fundamental equations of electrostatic, i.e.

∇.D=ρ

Next, we will see how we need to modify the equations to account for dielectric materials, which give rise to the quantity, ε_r, called relative permittivity.

Finally, we look at the experiments concerning current-carrying conductors, which will give us the second equation for electrostatic, i.e.

∇xE=0;

This is all there is to electrostatic! The rest are just clever mathematical tricks performed on the equations to solve complex problems.

1. Fundamental electromagnetic field and wave theory

1. Introduction

1.1 Prologue

I have read quite a number of textbooks on the subject of electromagnetism and my favourite ones are 'Electromagnetics for Engineer' by Ulaby and 'Field and Wave Electromagnetics' by D. K. Cheng. Both textbooks have pros and cons in the way they presented this subject to the public but they complement each other well.

The book by Ulaby has a more practical approach and therefore emphasized the applied side of electromagnetism. This is usually more suitable for the general engineers - you don't ask (too much) why, as long as it works. The book by Cheng is more mathematically rigorous and it could be quite tough for students without strong mathematical foundations. However, it looks at the Maxwell's equations as a mathematical subject, which is what it should be. At higher levels, I reckon, students will appreciate Cheng's book more. Nevertheless, both are good introductory books to electromagnetism and I highly recommend them for undergraduate students.

I have always wanted to write a book on Electromagnetism myself. And I will attempt to do so here in my blog. I hope to distinguish my little online 'book' on electromagnetism from the other textbooks by trying to explain the world of electromagnetism in as little jargon and mathematics as possible. This is no doubt daunting because mathematics is behind everything in Maxwell's equations.

Also, most of the textbooks chose to explain the equations first, then the observations (as a consequence of the equations). To explain electromagnetism in this order is understandable, because it would be easier. However, giving explanation in that order will give little information on how the equations itself emerged and creates a false impression that the equations emerged before the observations. This 'book' will attempt to cover this gap.


1.2 Objective

My online 'book' would, of course, be unable to replace the actual textbooks. I am obviously unable to give a detail account of the subject using many mathematical equations and graphs as the textbooks are able to. But what I will attempt to do here is to provide an account of the subject from my perspective and attempt to explain the theories of electromagnetism in a layman manner. I feel that many of the textbooks explained electromagnetism from a very mathematical point of view, and many undergraduate students ended up memorising Maxwell's equations without really knowing what they mean physically. Upon completion of a course in electromagnetism, the students can apply Stoke's and Gauss's theorem effortlessly and know that the divergence of electric flux density equals to the total charge density. But what does all these means in the physical world? This will usually elude the general students. I hope this book will be able to give more insight into the physical meaning of the governing equations of electromagnetism.

Also, there is a need for students to know that the process of scientific inference began not from equations, but from experimental observations, i.e. experiments are carried out in order to find the laws of the nature and mathematical models are proposed to explain the observations of the experiments. Not the other way around. All modern scientific theories take a similar path. First, there is the law of nature. Then, there are the observations. Mathematical models are proposed to explain the observations. Finally, the models are used to predict other results or to engineer new products.


Therefore, for example, to explain why one of the Maxwell's equations says that the divergence of magnetic flux is equal to zero? It is simply because from experimental observations, magnetic flux always close upon themselves. In mathematical form, it is denoted as '∇.B=0' . There is no point further asking why to this, because there is really no reason apart from the fact that no other experimental observations deviated from this mathematical equation (of course, this changed with the emergence of quantum mechanics, but we shall leave that to another time).

The equations make more sense if you look at it as a 'model' to explain the experimental results. Therefore, in order to fully appreciate Maxwell's equation it is necessary to understand the major experimental observations from late 18th century and follow their historical development that led to Maxwell's famous equations. As you may have guessed now, Maxwell did not perform the experiments himself. He merely came up with the 4 equations that succinctly describe all the phenomenon observed in the experiments. The word 'merely' may be an understatement here, because this was not an ordinary feat.

It is worthwhile to point out that with the emergence of quantum mechanics in the early 20th century, the Maxwell's view of electromagnetism is not entirely accurate, especially at the atomic level. This is why some textbooks emphasize the word 'field theory' when they explain electromagnetism using Maxwell's equations. As we shall see later, Maxwell's equations worked on the assumption that electric charges produced 'invisible' fields into the space and exerted forces along these field lines. Nevertheless, on the macroscopic level (up to about 100nm), Maxwell's equations work perfectly well.


1.3 Pre-requisite

One of the most annoying part of being an undergraduate student is finding a suitable textbook. Some books are too difficult and some books are just too easy. There's never a book that is 'just nice'! And I'm sure, as students, we constantly find ourselves in the situation of reading a textbook only to find that it contained some weird symbol or equation that you have never came across before. You will have to pick up another few books, just so that you could understand them. Only to realise that the few other books that you picked up, also contained some symbol or equation that you do not understand. And the search goes on...

With the advent of internet, things become much easier. But nevertheless annoying. Therefore, it is important to write a book that is self-containing and has a clear pre-requisite in order for the reader to fully appreciate it.

For this book, I assume that the reader would have a sound understanding of A-level physics and elementary mathematics (further mathematics not required). The reader would also be familiar with basic matrix and vector operation as well as some knowledge in vector calculus. All other equations used in this book will be derived from this basic understanding or other equations that have been derived earlier in the book. Therefore, with this pre-requisites, I hope this book will be self-contained.


1.4 Structure

I shall structure the book into the following chapters: chapter 1 is introduction (this chapter); chapter 2 on electrostatic; chapter 3 on magnetostatic; chapter 4 on ... (I will add/modify this as I write more). I will put a tag called 'EM book' on all the entries so that when you filtered my blog using this tag, you will get the 'book' in the order of the chapters described here.

These chapters shall be written in a modular format so that readers can jump to a particular chapter without any loss of continuity if they wish to.

Sometimes, I will digress from the main topic to provide additional detail. Whenever this happens, I will denote it with a long asterisk line like this:

*********************************
digressing from main topic
and additional detail here
but you can choose to skip
*********************************

In order for the reader to have a continuous flow of ideas, the reader could choose to skip those paragraphs contained within the two asterisk lines.


The book begins now... and if you like what I am doing, tell your other engineering undergraduate friends and drop me a comment or two so that I know.