Unit 1 - Field lines

How do we convey the information that is our knowledge of a field? We can state a formula, but to most people that doesn't help much in grasping the overall picture. (It is sometimes quite enough for a limited problem, but that depends on the problem.) We can try to use words, but that is usually even less helpful. For a scalar field, we can use a 'contour' map; the colored map in some newspapers is a contour map for the forecast temperature field. In the case of a vector field, a contour map won't be enough; we need direction as well as size at each point. We can put little arrows at various points to show direction; this is done to show the wind field on some weather maps.

However, there is one visual method that has been generally adopted for the vector fields that we use in physics. It can be used for any vector field; school children frequently get a taste of this method in connection with magnetic fields, hydraulic engineers and others use it to represent fluid velocity fields, etc. This is the method of using field lines.

There are two properties that are used to construct field line diagrams; those properties also let a viewer, even a beginner, understand them almost at a glance:
First, at each point on a field line, the direction of the field line is the direction of the corresponding field at that point. Thus, a diagram with arrows could become a field-line diagram by just smoothly connecting the arrows.
Second, in each region, the density of the field lines is proportional to the strength of the field there. Where the field magnitude is large, there are many field lines; where the field is weak, there are few field lines. The exact proportionality is at the discretion of the person preparing the drawing, and can be affected by his patience or the capability of his equipment, for instance.

There are some other properties that students are used to seeing in field lines; some of these properties are necessary consequences of the definitions; others are not.
For instance, field lines (for the same field) can never cross. (If they did, the field at the intersection would be a vector with two directions.)
Also, electric field lines point away from positive charges, toward negative charges (provided the charges in question are the dominant contribution to the field, which is not always the case).

But the property that electric field lines begin on positive charges and end on negative ones does not follow from the definition of field lines. Instead, it arises from a match between the dimension of our physical space, which means that the surface area of a sphere increases as the square of the distance from the center; and the distance dependence in Coulomb's Law, that the strength of the electric field decreases as the square of the distance from the charge. Hence if a fixed number of lines originates at a charge, then as they continue away they have to spread out, and their density falls as the area they spread to fill increases.

Contrast that with this hypothetical example. We invent a "charge" obeying an inverse cube force law. We place one charge at the origin, and suppose that we present its field lines on a scale such that the line density on a spherical surface 1 m away from the from the charge is 10 lines/cm². Since the area of that surface is 4p m², there are 4000p, or 12.6 thousand, lines crossing that surface. Now check the situation at 2 m away. Since the field is inverse cube and we are twice as far out and the line density is proportional to field strength, therefore there are only 1.25 lines/cm² here. There are more square centimeters here, but only four times as many, so the number of lines crossing the sphere is only 2000p or 6.28 thousand. Where did the other 6.28 thousand lines go? They had to end, and they did, but they couldn't end on charges because there weren't any.

(Actually, the quantum field theory description of electric phenomena suggests there may be more than a coincidence here, that Coulomb's Law is inverse square because space is three-dimensional.)

In any case, since electric field lines can begin or end only on charges, we can learn something about the number of charges by simply counting field lines. This is the content of Gauss's Law, the next topic. E-M Home Page/Unit 1/

Format for this page adapted with permission from pages
constructed by Dr. Lionel D. Hewett for his course
Modern Physics 1.