A picture to represent a physical law

There would be an extra advantage if one could describe dia-gramatically the physical laws which govern a field. Figure 10.1 may describe which way the wind is blowing or map out the strength of currents in a body of water. It does not give information on the gas laws or on the laws of hydrodynamics. What we need is a picture which not only describes the features of the field but also tells about the physical laws which govern the field and can perhaps be used to predict consequences of these laws.

A representation of electric fields which incorporates the conditions imposed by Coulomb's law is generally credited to Michael Faraday (1791-1867). His idea was slightly different from the representation shown in Figure 10.1, in that he drew

Figure 10.1 A vector field.

hypothetical 'lines of force' instead of vectors. The tangent to the line of force at any point indicated the direction of the field at that point. The density of the lines indicated the strength of the field.

Strong field

Weak field

Faraday's convention represents the strength of the field by the density of the lines of force.

The simplest electric field, that due to a single positive electric charge, is illustrated in Figure 10.2. The lines of force spread out symmetrically. All lines start at the charge, and once we move away from the charge, no lines are created or destroyed. It follows that if we draw an imaginary sphere around the charge, every line of force will cross that sphere. Surprisingly, this simple and obvious observation leads to important physical consequences.

Faraday's convention represents the strength of the field by the density of the lines of force.

Figure 10.2 Electric field due to a single positive point charge.
Michael Faraday

We make the rule that the number of lines leaving the source charge is proportional to the magnitude of the charge. By convention the number of lines emerging from a charge q is chosen to be q/e0. This means that 1.13 x 1011 lines leave a charge of one unit charge (1 coulomb).

There is an advantage which arises quite naturally in Faraday's method. The surface area of our imaginary sphere increases in proportion to the square of the radius

as the sphere gets bigger.* Therefore, as we move away from the source, the density of the lines of force crossing the spherical surface decreases in proportion to 1/(radius)2. If no new lines appear, and no lines disappear in space, the picture tells us that the force field obeys the inverse square law! The above argument is not limited to the choice of a spherical surface. If we surround the charge by any closed surface all lines will cross it sooner or later (some might cross and re-enter, but eventually will leave again).

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