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Charge and Electric Fields

1.1 Charge as a Fundamental Property of Matter

All matter is comprised of fundamental sub-atomic particles which themselves have basic properties, such as mass, charge and spin. The sub-atomic particle which electrical engineers are probably most concerned with is the electron, which was discovered by the Nobel Prize winning physicist J.J. Thomson in Cambridge in 1897 (Thomson 1897). Among its fundamental properties, electrons have a mass of 9.109×10-31?kg and a charge of -1.602×10-19?C, the magnitude of which we call e. Of these two properties, we are probably more familiar with the concept of mass as the world around us is dominated by gravitational forces at the macroscopic scale. We understand that two particles which have non-zero mass both experience a force between them. We rationalize this by saying that a particle with mass produces a gravitational 'field' which extends spatially away from the particle. Another particle with a non-zero mass inside this field then experiences a force.

As engineers, we do not tend to worry about the exact mechanism by which this force is being exerted over some distance, leaving that important consideration to physicists. Instead, we simply apply the equations for force between masses, such as that exerted by the earth on all structures in civil engineering. Therefore, we should be content to accept the less familiar concept of charge on the same basis: a particle with a non-zero charge produces an electric field which extends spatially away from the particle, and another particle with a non-zero charge inside this field then experiences a force. We use the symbol Q or q to denote charge, and the SI unit of charge is the coulomb (C).

1.2 Electric Field and Flux

Fields are widely used in physics to describe regions of space in which an object with a particular property experiences a force. Therefore, an electric field is a region of space in which an object with charge q experiences a force F. As force is a vector quantity, having both a magnitude and direction, electric field must also be a vector quantity E, so that

From this equation, it can be seen that the unit of electric field is N C-1, although, as we will see in Section 1.3, the more common unit is V?m-1.

In everyday life we experience objects with both positive and negative charge, because while electrons have a charge of -e, protons, which are one of the sub-atomic constituents of the nucleus of atoms, have a positive charge of e. Therefore, the force acting on a positive charge at a particular point in an electric field will act in the opposite direction to the force on a negative charge at the same point. This is the reason why like charges repel each other whereas opposite charges attract. This is in contrast to mass, which is positive for all matter, and therefore the force between masses is always attractive.

To assist us in visualizing fields, we use the concept of flux. We imagine that the electric field is composed of lines of flux whose direction at a given point in space is the direction of the electric field at that point and whose number density per unit area relates to the magnitude (or intensity) of the electric field.

We know that charge produces electric fields, and therefore lines of electric flux begin on positive charges and end on negative charges. As the total sum of all charge in the universe is zero, it must be the case that every line of flux that begins on a positive charge must have a balancing negative charge somewhere to end on. We can now visualize the electric field around a small point charge +q in free space (a vacuum) in Figure 1.1. If we assume that the balancing charge of -q is uniformly distributed an infinite distance away, then lines of electric flux will radiate uniformly away from the point charge. This will cause the field to decrease with increasing radial distance r from the point charge as 1/r2, just as we find for gravitational fields around mass as well. This is the basis behind the Coulomb law for the magnitude of the force F that acts on a charge q2 in an electric field of magnitude E1 produced by another point charge q1:

where r is the distance between the charges.

In Eq. (1.2) we have had to introduce a new quantity e0, which is the permittivity of free space. It is a fundamental constant with the value 8.854×10-12?F?m-1, and it is required to yield a result for the force between two charges that is correct in SI units.

Figure 1.1 Lines of electric flux around a point charge +q in free space assuming that the balancing charge is uniformly distributed an infinite distance away. The direction and area density of the flux lines are a measure of the electric field E at any point.

1.3 Electrical Potential

Let us imagine that we have two point charges of equal magnitude but opposite sign, +q and -q, which initially exist at the same point in free space, so that there is no electric field around the charges. If we slowly move the charge -q away from the +q charge so that the distance r between them is increasing, then an electric field distribution will be created around and between the charges as shown in Figure 1.2. As there will be a force acting on the charge that is being moved, given by Eq. (1.2), there must be some change in energy - work is being done against the force that is attracting the charges together. In this case, mechanical energy is being converted into an electrical potential energy, which could be converted back into mechanical energy again by allowing the two charges to accelerate back towards each other once more. It is a key concept that whenever a field (whether electric, magnetic or gravitational) occupies a volume of space, then some potential energy has been stored.

We can use basic mechanics to relate electric field to potential energy. If we have a charge q, in an electric field of magnitude E, which is moved by a small distance dx in the direction of the electric field, then there will be a change in potential energy of the charge given by

where F is the magnitude of the force acting on the charge due to the electric field. The change in potential energy is negative as the force acting on the charge is in the same direction as the movement. We define a new quantity, the potential difference, which is the change in energy per unit charge between two points in space. The potential difference is given the symbol V and has units of volts. Therefore, the small potential difference dV between the two points separated by the distance dx over which we have moved our charge q is

Equating dW in Eq. (1.3) and (1.4) gives

and therefore, by basic calculus, we have the result that

In other words, the electric field is the negative of the potential gradient. For readers who are familiar with vector calculus, we can rewrite this in three dimensions as

It should be noted that we can only ever talk about a potential difference between two points, for example the potential at a point a with respect to a point b which we could denote Vab. The direction is significant, as Vba = ?-?Vab. We often use arrows to denote a potential difference where we are considering the potential at the tip of the arrow with respect to the tail. If we know the electric field distribution between the two points, then we can evaluate this. A simple integration of the two sides of Eq. (1.6) would suggest that

Figure 1.2 The electric field distribution around two point charges +q (right) and -q (left) separated by increasing distance from (a) to (c).

However, this is a slight simplification as we know that the electric field is actually a vector quantity, and it is therefore only the component of a small movement dx parallel to the direction of the electric field that will lead to a change in potential. If we use vectors to express both the electric field E and the small movement dx, then the scalar (dot) product of the two yields exactly this result. Therefore, Eq. (1.8) is more generally expressed...