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.