Maxwell's Equations, Part V

Jan 01, 2012
Volume 27, Issue 1

This is the fifth installment in a series devoted to explaining Maxwell's equations, the four mathematical statements upon which the classical theory of electromagnetic fields — and light — is based. Previous installments can be found on Spectroscopy's website ( Maxwell's equations are expressed in the language of vector calculus, so a significant part of previous columns have been dedicated to explaining vector calculus, not spectroscopy. For better or worse, that's par for the course, and it's my job to explain it well — I trust readers will let me know if I fail! The old adage "the truth will set you free" might be better stated, for our purposes, as "math will set you free." And that's the truth.

Figure 31: The "shape" of a magnetic field about a wire with a current running through it.
In mid-1820, Danish physicist Hans Oersted discovered that a current in a wire can affect the magnetic needle of a compass. These experiments were quickly confirmed by François Arago and, more exhaustively, André Marie Ampère. Ampère's work, which defined a so-called "magnetic field" (labeled B in Figure 31), demonstrated that the effects generated were centered on the wire, perpendicular to the wire, and circularly symmetric about the wire. By convention, the vector component of the field had a direction given by the right hand rule: If the thumb of the right hand were pointing in the direction of the current, the curve of the fingers on the right hand gives the direction of the vector field.

Other careful experiments by Jean-Baptiste Biot and Félix Savart established that the strength of the magnetic field was directly related to the current I in the wire and inversely related to the radial distance from the wire r. Thus, we have

where "∝" means "proportional to". To make a proportionality an equality, we introduce a proportionality constant. However, because of the axial symmetry of the field, we typically include a factor of 2π (the radian angle of a circle) in the denominator of any arbitrary proportionality constant. As such, our new equation is

Figure 32: A wire loop generates a magnetic field B when a current I runs through the wire. In this case, the magnetic field is an axial field about the central axis of the loop.
where the constant µ is our proportionality constant and is called the permeability of the medium the magnetic field is in. In a vacuum, the permeability is labeled µ0 and, because of how the units of B and I are defined, is equal to exactly 4π × 10-7 tesla-meters per ampere (T∙m/A).

Not long after the initial demonstrations, Ampère had another idea: curve the wire into a circle. Sure enough, inside the circle, the magnetic field increased in strength as the concentric circles of the magnetic field overlapped on the inside (Figure 32). Biot and Savart found that the magnetic field B created by the loop was related to the current I in the loop and the radius of the loop R:

Figure 33: A magnet inside a coil of wire (top) does not generate a current. A magnet moving through a coil of wire (bottom) does generate a current.
Multiple loops can be joined in sequence to increase B, and in 1824–1825 English inventor William Sturgeon wrapped loops around a piece of iron, creating the first electromagnet. Even by then, Ampère had the thought that it was the current — that is, moving charges — that caused the magnetic field.

Joseph Henry was an American scientist who eventually became the first secretary of the Smithsonian Institution. In 1830, he performed some experiments showing how a magnetic field can induce electricity and did not publish it. Because of this, he lost a larger place in scientific history when in 1831, Michael Faraday announced that a changing magnetic field could produce an electrical current. (Henry's work has not gone unnoticed, as the SI unit of inductance is named the henry.) Note that Faraday (followed by others) found that a changing magnetic field is required; a static, nonchanging magnetic field produces no current (Figure 33). This strongly suggests that an electric current I is related to a varying magnetic field, or

Actually, this is not far from the truth (which would then be another of Maxwell's equations if it were), but the more complete truth is expressed in a different, more applicable form.

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