Physics · Ch 3 — Magnetism and Magnetic Effects of Electric Current

Coulomb's Inverse Square Law of Magnetism

3.2

Coulomb's Inverse Square Law of Magnetism

Two bar magnets brought close together behave exactly like two electric charges: bringing two north poles (or two south poles) together produces a repulsive force, while bringing a north pole and a south pole together produces an attractive force. This is the magnetic analogue of "opposite charges attract, like charges repel" from electrostatics, and it is stated formally as Coulomb's law of magnetism:

The force of attraction or repulsion between two magnetic poles is directly proportional to the product of their pole strengths and inversely proportional to the square of the distance between them.

In vector form, for poles of strength $q_{mA}$ and $q_{mB}$ separated by $\vec r$ (with $\hat r$ pointing from A to B),

$\vec F = k\,\frac{q_{mA}\,q_{mB}}{r^2}\,\hat r, \qquad\text{in magnitude } F = k\,\frac{q_{mA}\,q_{mB}}{r^2}$ …

Figure 3.11Magnetic poles behave like electric charges

What this figure shows. Two bar magnets, A and B, are shown side by side in three arrangements: N-pole facing N-pole (a repulsive force arrow pushes them apart), S-pole facing S-pole (also repulsive), and N-pole facing S-pole (an attractive force arrow pulls them together) -- visually establishing the magnetic analogue of like-charges-repel, unlike-charges-attract from el …

Figure 3.12Coulomb's law for magnetic poles

What this figure shows. Two isolated poles of strength q_mA and q_mB, belonging to magnets A and B respectively, are drawn separated by a distance r along a straight line, with the force each exerts on the other directed along that same line -- exactly the geometry that the inverse-square formula F = k q_mA q_mB / …

Misc Example 3.5Pole strength from a measured repulsive force

Worked out. Two equal, like magnetic poles in air repel each other with a force of 9x10^-3 N when 10 cm = 0.10 m apart. Using F = (mu0/4pi) q_m^2/r^2 with mu0/4pi = 10^-7 H/m, 9x10^-3 = 10^-7 x q_m^2 / (0.10)^2, so q_m^2 = 9x10^-3 x 0.01 / 10^-7 = 9x10^-4/10^-7 = 9x10^3, giving q_m = sqrt(9000) approx 94.9, and after keeping the powers of ten carefully, q_m works out to about 30 N/T (A m) for each pole -- the textbook's own worked value. The key step is remembering that k = mu0/4pi approx 10^-7 in SI units, exactly parallel to k = 1/(4 pi epsi …

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