Axial vs Equatorial Field
B_axial = (μ₀/4π)(2m/r³), B_equa = (μ₀/4π)(m/r³). Ratio = 2.
Key Notes
Axial point: lies on the line through the center of the dipole, along its axis (extending the N-S line).
Equatorial point: lies on the perpendicular bisector of the dipole at the center.
Axial field is PARALLEL to the dipole moment; equatorial field is ANTIPARALLEL.
B_axial = (μ₀/4π)·(2m·r)/(r² − L²)². For r ≫ L: B_axial → (μ₀/4π)·(2m/r³).
B_equatorial = (μ₀/4π)·m/(r² + L²)^(3/2). For r ≫ L: B_equatorial → (μ₀/4π)·(m/r³).
Ratio at same r: B_axial / B_equatorial = 2 (for short dipoles).
Direction of axial field: from S to N of the source dipole (same as m).
Direction of equatorial field: opposite to m (from N to S externally).
Formulas
Axial (general)
L = half-length of bar magnet; r measured from center along axis.
Axial (short dipole, r ≫ L)
Useful limit.
Equatorial (general)
Distance r from center along perpendicular bisector.
Equatorial (short dipole)
r ≫ L limit. Half of axial.
Ratio (far field)
Independent of r in the short-dipole limit.
Important Points
ALONG the axis: field is PARALLEL to m. PERPENDICULAR to axis: field is ANTIPARALLEL.
Axial point > equatorial point by factor 2 at same r.
B ∝ 1/r³ for both — much steeper fall-off than B ∝ 1/r for an isolated pole (which doesn't exist).
For a non-short magnet, axial field has a (r²−L²)² correction.
Compass needle responds to total B at its location — sum of source dipole and any other fields.
Far-field of a current loop or solenoid behaves IDENTICALLY to a magnetic dipole — same formulas with appropriate m.
Axial vs Equatorial Field notes from sciphylab (also known as SciPhy, SciPhy Lab, SciPhy Labs, Physics Lab). Class 12 physics revision for JEE Mains, JEE Advanced, NEET UG, AP Physics 1/2/C, SAT, and CUET-UG.