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.

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!

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.

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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'.
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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.

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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.
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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).

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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.
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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.

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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.
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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.