Home  18.013A  Chapter 28 


The systematic study of electrical and magnetic forces began in the late Eighteenth Century. Electrically charged objects were produced by rubbing one substance with another. Two kinds of charge were observed, like charges repel one another, and unlike charges attract.
Benjamin Franklin performed experiments with electrically charged pith balls, which led Priestley and Cavendish to try to prove that the electric force was an inverse square force just like the gravitational force. Coulomb proved by his experiments that the force between two unlike charges is indeed inverse to the square of their distance apart, and along the line joining them. The attraction has exactly the form of gravitational attraction between masses.
At some stage someone got the idea of describing electrical forces through the notion of an electric field. This is defined as the force per unit charge that a small fictitious charged object would experience from a given distribution of charges when placed at the argument of the field.
The surface area of a sphere is proportional to its radius squared, while the electric field of a point charge is inversely proportional to that same quantity. Therefore the integral of the normal component of an electric field around a spherical surface is independent of the radius of the sphere, and only measures the strength of the charge at its center.
We call the integral of the normal component of a vector $\stackrel{\u27f6}{W}$ over any surface the "flux of $\stackrel{\u27f6}{W}$ through the surface". The remark above can be stated as: the flux of electric field through the surface of a sphere containing a charge at its center is proportional only to the amount of charge and is a constant times the amount of that charge.
Gauss generalized this statement to apply to the surface of any region containing the charge by means of the divergence theorem, which he discovered. This theorem implies that the flux of electric field through the boundary of any region of space is an appropriate constant multiplied by the amount of electric charge in the region.
Around the turn of the Nineteenth Century, Volta invented the battery, and it became practical for people to produce currents of electricity. Oersted and Ampere discovered in about 1820 that electric currents produce forces that cause magnetized needles to line up in a direction tangential to a circle about the wire. Ampere, in particular, discovered that the magnetic force on such needles produced by a long straight wire carrying electric current is proportional to the current flow and inversely proportional to the distance of the needle from the wire.
The circumference of a circle is proportional to its radius and the magnetic field just described is inversely proportional to radius. Therefore the integral around the circumference of any circle around the wire of the component of magnetic field in the direction of the path is independent of its radius. It is an appropriate constant times the "flux" or flow of current through the wire for any such circle.
We define the integral of the component of a vector field $\stackrel{\u27f6}{W}$ in the direction of a path around a closed path to be the circulation of $\stackrel{\u27f6}{W}$ around that path. Ampere's law can then be stated as: the circulation of magnetic field around a wire is a constant times the flux of current through it.
Faraday got the idea that if electric current flux causes magnetic circulation, then there should be some sort of reciprocity: magnetic flux ought to be able to cause electric current circulation. In 1831, after looking for such an effect, he discovered his celebrated law of induction: that changing magnetic flux through a surface $S$ produces a circulation of electric field on its boundary. This means it produces a "difference in electrical potential" around the boundary path of $S$ which means a charged particle in a wire around it will have work done on it in moving around the wire. This will make electric current flow in a wire around the surface, and that current is a constant times the derivative of the magnetic flux through (any) surface bounded by the wire.
By increasing and decreasing the amount of current in one wire, you can make its magnetic force oscillate, which will cause current to flow back and forth in another wire.
Somewhere in the middle of the century Stokes discovered his mathematical theorem relating flux of a curl a vector field $\stackrel{\u27f6}{W}$ on a surface $S$ to the circulation of $\stackrel{\u27f6}{W}$ around the boundary of $S$ .
Maxwell used this fact to prove that consistency of the equations of electricity and magnetism requires a modification of Ampere's Law when there are changing electric fields.
With this modification he noted (in circa 1862) that electric and magnetic fields can display wavelike behavior even in the absence of matter, and he asserted that the phenomenon of light consisted of exactly such waves. His claims produce a prediction of the velocity of light, which had only recently been measured, and it agreed with that measurement precisely.
His celebrated differential equations describing the behavior of electromagnetic fields were published in 1874. Maxwell's discoveries were distinguished by being entirely theoretical. He utilized the mathematical implications of Stokes' Theorem rather than an experiment to discover his "displacement current" whose presence made possible his identification of light with electromagnetism.
The idea behind Maxwell's discovery is this: according to Stokes' theorem, the flux of the curl of a vector field $\stackrel{\u27f6}{W}$ through a surface is its circulation around the boundary of the surface. By this theorem, the flux of the curl any vector field must be the same through any two surfaces with the same boundary.
Faraday's discovery allowed people to produce electric current that oscillates by moving magnets near wires, or (equivalently) by moving wires near magnets (as in Davis and Kidder's Therapeutic Device, patented in 1854).
Induction of current in one wire from the change in current in another assumes that the changing current in the first produces a changing magnetic field which produces the current in the second according to Faraday's law.
But now suppose we have a gap in the second wire. Current will flow in it until charge builds up across the gap, and this current will produce a magnetic field of its own. If the current in both wires is made to oscillate, that is, to flow back and forth as a sine function of time, current will flow much of the time despite the gap, and if the frequency of the sine is large enough, there will be little charge build up at the gap at any time, and little "impedance" to current flow in the wire.
According to Ampere's Law (applied to a nonsteady state current situation) there will be oscillating "magnetic circulation" around the wire, from the oscillating current flow in the wire in this situation. But if we take a surface that passes through the wire and deform it to make it pass through the gap instead of the wire, there will be no current through it! Then Ampere's law would say there was no magnetic circulation on the same path in contradiction to the previous statement.
A circle around the wire with a gap can be filled by a surface that passes through the wire or else by distorting that surface keeping its boundary the same, into one that goes only goes through the gap. The integral of flux of a curl must be the same for both surfaces. There must therefore be something in the gap that, like current, contributes to the flux of the curl of the magnetic field there.
Maxwell concluded that the current flux could not possibly be the flux of the curl of the magnetic field under these circumstances. Ampere's law, which describes steady state current flow adequately must be modified when current flow is time dependent!
The current flux with a given boundary will be different depending on whether we pass our surface through the wire or through the gap. The flux of the curl of the magnetic field must be the same in both. If the curl of the magnetic field is to be flux of current in the wire it must be something else in the gap and that something else must have the same flux.
The only thing we know about in the gap is that it contains the changing electric field caused by the charge oscillations on its faces. Maxwell postulated that consistency requires an additional time dependent term in Ampere's law proportional to the flux of the time derivative of this electric field.
This term, which he called "displacement current" produces remarkable symmetry in the resulting equations. When written as differential equations, the laws of Gauss, Ampere with Maxwell's modification, and Faraday have the consequence that electric and magnetic fields obey "the wave equation" in the absence of matter, and suggest that there can be waves of electric and magnetic field, which waves move with a finite velocity.
Maxwell's assertion that light is a form of such wave motion implies a particular finite velocity of light that can be deduced from electric and magnetic phenomena. The symmetries of his equations include not only rotations in ordinary space, but also transformations which mix space with time, called Lorentz transformations.
In 1888, Hertz actually created electromagnetic waves and detected them in his laboratory. He had connected a coil of wire and small gap, ran current through the wire until the field on the condenser caused a spark; the resulting oscillations of current produced waves that were observable on another similar circuit.
Marconi got the notion from this that such waves could be used for communication by causing current to flow in a distant wire. In the 1890's he set up apparatus to transmit signals over ever widening distances, and by 1901was able to send telegraphic signals across the Atlantic Ocean that could be and were received and used for wireless communication.
The physical laws involved in these subjects are few in number and can be stated in a few lines. We will now consider their mathematical implications in terms of the concepts of vector calculus.
