Magnetic pole strength

Magnetic pole strength (symbol: p) is a physical quantity used to measure the strength of the pole of a bar magnet (or a hypothetical magnetic monopole). If there is an infinitely long wire where the electric current is I, then the magnetic pole strength is defined as follows:

$p = \frac{W}{I}$,

where W is the work which has been made when the magnetic pole has been turned around the wire in a closed loop. In the definition, it has also been determined that p > 0 for the north magnetic pole and p < 0 for the south magnetic pole.

The Ampere law follows almost immediately from the definition. The notion of magnetic pole strength may seem artificial, but when it is defined as above, it is perfectly analogous to the electric charge. Thus the magnetic field strength is defined just as the electric field strength: as the force with which the field affects to a unit north pole:

$\mathbf{H} = \frac{\mathbf{F}}{p}$.

In the International system of units (SI) there is no official standard unit for magnetic pole strength, but the unit of magnetic flux, 1 weber (Wb) = 1 J/A = 1 Vs, can also be used for magnetic field strength. The magnetic flux around the magnetic pole is equal to the magnetic pole strength, just as the electric flux around a charge is equal to the charge.

Alternatively the magnetic pole strength is sometimes defined as the magnetic moment of a magnet divided by the distance of its poles. In this case, its unit us one ampere metre (1 Am). This quantity has not the same dimension as the one defined earlier, but their ratio is constant and equal to the universal magnetic constant, $mu_o = 4 \pi \times 10^-7 Vs/Am$.

Despite intensive research, no magnetic monopoles have hitherto been found, but in the modern particle physics such particles are considered to be theoretically possible. A bar magnet, however, especially if it is long and narrow (for example the needle of a compass), behaves very much as if it were composed of two magnetic monopoles being in both ends of the bars, having equal strength but opposite sign.

Magnetic Coulomb's law

A bar magnet can be analyzed as if it where composed of two magnetic poles of opposite sign, being at its opposite ends. In this case it turns out that the poles of two distinct magnets affect to each other with a force being directly proportional to the product of pole strengths and inversely proportional to the square of their distance. This so called magnetic Coulomb's law is exactly analogous to the electrical Coulomb's law, and it can be written as a formula:

$F = k \frac{|q_1 q_2|}{r^2}$

where k is a constant. Formerly the cgs system of units was very much used in physics. In it, the electromagnetic units were defined so that this constant was one. Thus the unit of magnetic pole strength and of magnetic flux, one maxwell, was equal to 10⁻⁸ Wb. Such a magnetic pole affects on another equal pole, being at the distance of one centimeter, with the force of one dyne.

In current SI units, the constant k must be expressed by using the magnetic constant or the permeability of vacuum, μ₀ avulla. It follows from the definition of the ampere that this constant is equal to 4 π · 10⁻⁷ Vs/Am. If the weber (or the volt second) is used as the unit of magnetic pole strength, the coefficient k in the formula above is equal to 1 / (4 π · μ₀). If the unit ampere meter is used instead, the coefficient is μ₀ / (4 π), which is exactly 10⁷ Vs/Am.

The magnetic Coulomb's law has mainly science-historical significance. Its relevance is limited by the fact that in most magnets, the location of poles cannot be precisely specified, and therefore magnetic pole strength cannot be measured with good precision. For this reason, magnetic moment has turned out to be a much more useful quantity and it can be measured more accurately. If, however, magnetic monopoles turned out to exist, then, according to theory, they would obey precisely this law.

See also

• magnetic monopole
• magnetic flux
• magnetic flux density

Sources

• Mansfield and O'Sullivan, Understanding Physics. Sons Wiley Ltd, 1998.

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