Field Lines of Static Charges

1

While we are still introducing some of the basic concepts of electromagnetism, we might extend this initial discussion of fields to include field lines and equipotential lines. The diagram opposite shows a single negative charge with lines that both connect to the charge and surround it. These lines are described as the field lines and the equipotential lines of the charge, although in practice neither type may actually exist with respect to a single charge in isolation.  

[1]      1

Examination of [1a] highlights that the definition of electric field [E] stems from Coulomb’s force law, such that it represents the force per charge [q], where [q] might be thought of as a  unit charge required to measure the strength of charge [Q]. So while [1b] might appear to define [E] in terms of [Q] only, in reality, another charge is also required, not only to measure [E], but to essentially instantiate the existence and the direction of field lines. As such, the force between one of more charges might be thought to alter the space around the charge by generating an electric field [E]. Any other charge placed in this field, e.g. the unit charge, will then experience a force [FE]. Alternatively, we could describe the [E] field in terms of a mechanism that is able to transport a force between 2 points in space.

In this initial discussion of field lines, we are primarily focusing on static configurations of charges. However, should a charge change its position in space, equation [1] suggests that the E-field force would also change in the surrounding space. However, this change would not be an instantaneous process as there is an inherent delay due to an upper limit of the propagation velocity [c] in vacuum. Later, we will discuss the implications of charges moving with constant velocity and acceleration.

However, given that so many EM phenomena are now explained in terms of electric fields, rather than just a charge exerting force on another charge through empty space; the E-field is now regarded as having a physical existence rather than being just a mathematical entity. In this context, the physical existence of the E-field is then extended into field lines, but these lines are still just a representation of the vector force, i.e. its strength and direction, at any given point in space. As such, one of the fundamental rules of drawing field lines is that the E-field vector direction must form a tangent to the field line at any given point; while remembering that the strength and direction of the vector can only be measured with respect to another charge, e.g. a unit charge probe. based on these assumptions, the electric field and the concept of lines of electric force can be used to map the forces per charge [q] created by charge [Q] over a region of space. 

2

The diagram above illustrates the field lines of force associated with the electric field between charges, where (a) shows the field lines connecting a negative-positive charge pair, while (b) shows the field lines separating two positive charges. The main rules that define the behaviour and properties of these field lines are listed as follows:

  1. Electric field lines start on positive charges and end on negative charges.
  2. The density of electric field lines indicates the strength of the E field in a particular region. The field is stronger where the lines get closer together.
  3. Field-lines never cross and never merge.
  4. Field lines connect perpendicularly to the source and sink charges.
  5. The direction of the electric force at any point on the field line must form a tangent to the field line.

So, collectively, field lines can convey information about the strength and direction of the electric field, i.e. force per charge, surrounding a given configuration of charges, as illustrated in the diagram below.  The direction of the field at any given point can be determined by forming a tangent to the field line, i.e. rule-5.

  3

As such, a collection of field lines form a field line diagram or field map. As indicated, the direction is determined by the tangent to the field line; while the strength is determined by the density of the field lines. In the example above, the field lines are closest together on the right-hand side, so the vector field is strongest in this region, while in the middle the separation of the field lines suggests a weaker vector field. Of course, the diagram above is only presenting a 2-dimensional view of the vector that must exist in 3-dimensional space.

4

Obviously, it would become increasingly difficult to represent the 3-dimensional field lines when linked to multiple charges placed at various positions in 3-dimensional space. Although the rules, as outlined above, essentially remain the same, it should be remembered that the implied field strength suggested in a 2-dimensional diagram can be deceiving in that you might not be ‘seeing’ all the field lines. The actual number of lines entering or emerging from a charge is only proportional to the charge strength when shown in 3-dimensions.

Equi-potential Lines

The definition of electric field lines is not the only way to show how a charge affects the space around it. Another scheme introduces the idea of the electric potential lines to plot a scalar field, as opposed to a vector field as previously described using field lines. In the diagram below, a region of space around a group of charges shows the normal field lines using solid lines with directional arrows. In contrast, the equipotential lines are shown using dashed lines, although in 3-dimensions, these dashed lines would become equipotential surfaces.

5

The difference in electric potential between any two points, e.g. A and B, is defined as the work per unit charge required to move a unit positive charge from A to B against the electric force. In a static configuration, this work depends only on the locations of A and B, not the path traversed. Therefore, in principle, an equipotential or scalar map can be drawn by choosing a given point (A) in space and assigning its electric potential on an arbitrary scale, which then proceeds to assign every other point in space a potential based on the work per unit test charge to move a test charge from (A) to the next point. However, in practice, it is usual to measure potential with respect to a zero reference, e.g. ground. At this point, it might be useful to draw some parallels between the definition of the electric [FE] and gravitational [FG] forces plus the electric [UE] and gravitational [UG] potentials:

[2]      2

If we considered the electric potential energy [UE], as per [2], then we must define this potential as a function of the distance between [Q] and [q], just as the gravitational potential is a function of the distance between [M] and [m]. However, in the case of [2a], we can assign the gravitational force [FG] to the inertial force [F=ma], which leads to the equivalence of gravitational mass and inertial mass, whereby [g=a] can be expressed independently of [m]. However, this equivalence cannot be used directly in the case of electric potential energy, but we might follow the logic used to define the electric field [E=F/q], i.e. the force per charge, to define the ‘electric potential’ or ‘voltage’ as the electric potential energy [UE] per charge [q]. In this context, the electric potential or voltage [V] is also independent of [q], although its existence is still required. However, it should be noted, that the work per unit charge or electric potential difference [V] is only independent of the unit charge [q] provided the electric field does not vary in time.

Anyway, as indicated, electric potential is a scalar definition, which is not that informative if simply viewed as a collection of scalar values for all points in space. Therefore, it is convenient to connect points of equal potential such that you end up with equipotential lines, in 2-dimensions, or equi-potential surfaces, in 3-dimensions. If a unit charge [q] is moved perpendicular to an electric field line then no work is done against the electric force and the potential along this line of motion remains the same, i.e. it defines an equipotential line. Therefore, electric field lines and equipotential lines always cross at right angles.

Scalar and Vector Potentials

One of the biggest problems for anybody attempting to understand the subject of electromagnetism is the apparent scope to describe the fundamental processes involved in so many different ways using so many different quantities and so many forms of mathematical notion. In part, this whole section attempts to minimise this scope by restricting the discussions to a minimalist subset. However, some mention has been made to the idea of scalar potentials and vector field, so now might be as good a point to, at least, introduce the general concept of scalar and vector potentials. In the wider scope of electromagnetic radiation, i.e. electrodynamics as opposed to just electrostatics, it is not always convenient to work with the electric and magnetic fields directly. In this expanded context, it is often common to see the definition of a ‘scalar potential’ and ‘vector potential’ introduced, which we might initially describe in the following terms:

  • Scalar [φ] potentials are linked to E-fields
  • Vector [A] potentials are linked to B-fields

We have already introduced the basic concept of a scalar potential in the previous discussion of potential lines, i.e. a scalar potential is a measure of the voltage at any given point. It might be realised that we could develop the idea of equipotential lines and surfaces into contour maps, where the rate of change of the scalar potential aligns to the gradient of the scalar potential.  In contrast to field lines and equipotential lines, which are primarily linked to the definition of the electric [E] field, a ‘vector potential’ is primarily associated with the magnetic [B] field. As a vector quantity it also has both magnitude and a direction and, as such, the vector potential can help define the relationship of the magnetic [B] field with respect to the electric [E] field. However, as a generalisation, if the potentials are changing with time, then the relation between the potentials and the fields can become very complex, but the size of the [E] and [B] fields remain proportional to the rates of change of the potentials in space and time. For example, when a point charge in space moves, the scalar potential at some other point must change as a function of the charge strength and the inverse distance to the charge. However, this distance is not defined by where the charge is now; but rather where the charge was, in order to account for the propagation delay linked to the propagation velocity [c]. This is sometimes referred to as the ‘retarded potential’.  As alluded to in the previous diagram, a charge may also be an aggregation of multiple point charges and, in such cases, the scalar potential would have to be calculated by adding up the retarded potential of each point charge due to each offset distance. In terms of this very brief introduction, we might simply relate the vector potential to a current flow, where each component of the current, i.e. charge per second, creates a retarded vector potential, i.e. magnetic field, which is proportional to the current flow and inversely proportional to retarded distance, due to the propagation delay.

 However, the significance of time-dependence in electromagnetism may only become clearer after we discuss the issue of Maxwell’s time-dependent and time-independent equations in more details in the subsequent pages.