Amperes law

When Ampère heard of Oersted's discovery, he immediately returned to his work on electric currents, and devoted his attention to developing a mathematical theory of magnetic phenomena. Magnetic field lines surrounding a current are quite different from electric field lines, in that magnetic lines form a closed loop with no beginning and no end, whereas electric lines always begin and end on electric charges. As we saw above, there was no evidence for single isolated magnetic poles acting as sources of magnetic field lines, unlike electric charges which act as sources of electric field lines.

Ampère continued the argument in that, if isolated magnetic poles existed, one such pole, placed near a current-carrying wire, would experience a force which would drive it around the magnetic loop for ever. What is more, the force would be tangential, causing a continuous increase in angular velocity, which would continue for ever and be difficult to reconcile with the principle of conservation of energy!

A bar magnet, which of course does exist, will experience different forces on its north and south poles, depending on its orientation. Again, from energy conservation, one could form the hypothesis that no matter what its orientation, no mechanical work is done in bringing a magnet along any path completely around the current. The work done on the individual north and south poles must be equal and opposite.

Figure 10.13 Bringing a bar magnet completely around a current.

Since for any circular path B ^ 1/R and the path length the work done in bringing the north end of the magnet around the large circle is exactly cancelled by the work done by the south end around the smaller concentric path.

The work done for a complete lap around any circle is the same. Since the hypothesis is not confined to a circular path, it can be generalised by stating that the work done in bringing a (imaginary) unit magnetic pole around any closed loop is the same and is directly proportional to the current i. The proportionality constant is the magnetic constant ji0, which we have just met in the Biot-Savart law.

Numerous experimental tests have confirmed this hypothesis and can be stated now as a law written as follows:

The law can be extended to apply to a magnetic field created by any number of electric currents going in random directions. Imagine current-carrying wires criss-crossing through space in all directions. If itotal = i1 + i2 + i3 + i4 + ••• is

the algebraic sum of these currents passing through the loop, then

(Any currents which pass outside the loop do not contribute to the integral.)

This formula is similar to Gauss's law in electrostatics except that, instead of an arbitrary Gaussian surface surrounding electric charge, we draw an arbitrary Amperian loop, and instead of total enclosed charge inside the Gaussian surface, we have total current through the Amperian loop.

André Ampère. Courtesy of La Poste, Monaco.

In practice the most efficient way to create a magnetic field is to send a current i through a coil of many turns. The magnetic field thus created looks just like the field produced by a bar magnet, as can be seen from the pattern of iron filings in the diagram on the left.

Ampere's law was given a stronger footing by a mathematical theorem due to George Stokes, an Irish mathematician who held the post of Lucasian Professor of Mathematics at Cambridge from 1849. Stokes' theorem applies to all vector fields and equates a path integral called the circulation around any closed loop to an area integral of vector flux over any surface bounded by that loop. Applying Stokes' theorem to electromag-netism we get a relationship identical to Ampere's law.

George Stokes (1819-1903)
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