32-4 Magnets

32-4 Magnets#

Prompts

  • Describe Earth’s magnetic field as a dipole. Where is the north geomagnetic pole located, and why does a compass needle point toward geographic north?

  • What are field declination and field inclination? Given Earth’s horizontal component and inclination (dip) angle, walk me through finding the total magnitude of Earth’s magnetic field.

  • What are lodestones? How do they relate to the origin of compasses?

Lecture Notes#

Overview#

  • Earth ≈ magnetic dipole; field near surface like a bar magnet through the center

  • Dipole axis makes ~11.5° with rotation axis; south pole of dipole is in Northern Hemisphere (so “north magnetic pole” is really the dipole’s south pole)

  • Local field: specified by declination (angle from geographic north) and inclination (angle from horizontal)

Physical intuition

A compass needle points toward the geomagnetic north pole (off Greenland). Because the dipole’s south pole is in the Northern Hemisphere, field lines emerge in the Southern Hemisphere and reenter in the Northern Hemisphere.

Why a compass aligns with the field#

A compass needle is a magnetized needle—a magnetic dipole with moment \(\vec{\mu}\). (An unmagnetized needle has no dipole moment and would not align.) In an external field \(\vec{B}\), the orientation energy is

(212)#\[ U = -\vec{\mu} \cdot \vec{B} \]

The dipole minimizes \(U\) by aligning \(\vec{\mu}\) with \(\vec{B}\). Thus the needle points along the local field direction.

  • A compass (needle on vertical axis) measures the horizontal direction.

  • A dip meter (needle on horizontal axis) measures the inclination from horizontal.

Physical intuition

\(U\) is most negative when \(\vec{\mu}\) and \(\vec{B}\) are parallel. The system seeks minimum energy, so the dipole rotates to align with the field. This is the same principle for any magnetic dipole in a field (see 32-5 for atomic dipoles).

Field declination and inclination#

  • Field declination: angle (left or right) between geographic north and the horizontal component of \(\vec{B}\)

  • Field inclination (dip): angle (up or down) between horizontal and \(\vec{B}\)

Measurement: Compass gives declination; dip meter gives inclination. From horizontal component \(B_h\) and inclination \(\theta\):

(213)#\[ B = \frac{B_h}{\cos\theta} \]

Summary#

  • Earth approximates a magnetic dipole; geomagnetic north pole (dipole south pole) is in Northern Hemisphere

  • Compass principle: Magnetized needle (magnetic dipole) minimizes \(U = -\vec{\mu} \cdot \vec{B}\) by aligning with \(\vec{B}\)

  • Declination: horizontal angle from geographic north; inclination (dip): vertical angle from horizontal

  • Total field \(B = B_h/\cos\theta\) from horizontal component \(B_h\) and inclination \(\theta\)

  • Lodestones: naturally magnetized stones; early compasses used lodestones or magnetized iron