The Electrostatic Analog

The Electrostatic Analog

The key insight is that the magnetic field of a bar magnet at large distances behaves exactly like the electric field of an electric dipole. This allows us to translate all known results from electrostatics into magnetism by making a simple substitution.

The Replacement Rule:

To go from an electric dipole (dipole moment $\vec{p}$) to a magnetic dipole (magnetic moment $\vec{m}$), replace:

• $\vec{E} \rightarrow \vec{B}$
• $\vec{p} \rightarrow \vec{m}$
• $\frac{1}{4\pi\varepsilon_0} \rightarrow \frac{\mu_0}{4\pi}$

This works because the mathematical forms of the field equations are identical.

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Field Expressions for a Short Bar Magnet

For a bar magnet of length $l$, at distances $r \gg l$, the magnet behaves as a point dipole.

1. Equatorial Field ($\vec{B}_E$)

On the perpendicular bisector of the magnet (the equatorial line), the field is:

• The negative sign indicates the field direction is opposite to the magnetic moment $\vec{m}$.
• This is analogous to the equatorial field of an electric dipole: $\vec{E}_E = -\frac{1}{4\pi\varepsilon_0} \frac{\vec{p}}{r^3}$.

2. Axial Field ($\vec{B}_A$)

On the axis of the magnet (the line through its poles), the field is:

• The field is parallel to $\vec{m}$ and twice as strong as the equatorial field at the same distance.
• This is analogous to the axial field of an electric dipole: $\vec{E}_A = \frac{1}{4\pi\varepsilon_0} \frac{2\vec{p}}{r^3}$.
• This equation is the vector form of Eq. (5.1) from the chapter.

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Application: Equilibrium of Two Magnetic Dipoles (Example 5.2)

Consider two identical magnetic dipoles P (fixed at O) and Q (movable). The field of P at any point is given by the axial or equatorial formula above.

• Stable equilibrium occurs when $\vec{m}_Q$ is parallel to $\vec{B}_P$ (minimum energy).
• Unstable equilibrium occurs when $\vec{m}_Q$ is anti-parallel to $\vec{B}_P$ (maximum energy).

For the configurations in Figure 5.4: …