Magnetic dip, dip angle, or magnetic inclination is the angle made with the horizontal by Earth's magnetic field lines. This angle varies at different points on Earth's surface. Positive values of inclination indicate that the magnetic field of Earth is pointing downward, into Earth, at the point of measurement, and negative values indicate that it is pointing upward. The dip angle is in principle the angle made by the needle of a vertically held compass, though in practice ordinary compass needles may be weighted against dip or may be unable to move freely in the correct plane. The value can be measured more reliably with a special instrument typically known as a dip circle. Dip angle was discovered by the German engineer Georg Hartmann in 1544. A method of measuring it with a dip circle was described by Robert Norman in England in 1581.
Explanation
Magnetic dip results from the tendency of a magnet to align itself with lines of magnetic field. As Earth's magnetic field lines are not parallel to the surface, the north end of a compass needle will point upward in the Southern Hemisphere (negative dip) or downward in the Northern Hemisphere (positive dip). The range of dip is from -90 degrees (at the South Magnetic Pole) to +90 degrees (at the North Magnetic Pole). Contour lines along which the dip measured at Earth's surface is equal are referred to as isoclinic lines. The locus of the points having zero dip is called the magnetic equator or aclinic line.
Calculation for a given latitude The inclination I {\displaystyle I} is defined locally for the magnetic field due to Earth's core, and has a positive value if the field points below the horizontal (i.e. into Earth). Here we show how to determine the value of I {\displaystyle I} at a given latitude, following the treatment given by Fowler. Outside Earth's core we consider Maxwell's equations in a vacuum, ∇ × H c = 0 {\displaystyle \nabla \times {\textbf {H}}_{c}={\textbf {0}}} and ∇ ⋅ B c = 0 {\displaystyle \nabla \cdot {\textbf {B}}_{c}=0} where B c = μ 0 H c {\displaystyle {\textbf {B}}_{c}=\mu _{0}{\textbf {H}}_{c}} and the subscript c {\displaystyle c} denotes the core as the origin of these fields. The first means we can introduce the scalar potential ϕ c {\displaystyle \phi _{c}} such that H c = − ∇ ϕ c {\displaystyle {\textbf {H}}_{c}=-\nabla \phi _{c}} , while the second means the potential satisfies the Laplace equation ∇ 2 ϕ c = 0 {\displaystyle \nabla ^{2}\phi _{c}=0} . Solving to leading order gives the magnetic dipole potential
ϕ c = m ⋅ r 4 π r 3 {\displaystyle \phi _{c}={\frac {{\textbf {m}}\cdot {\textbf {r}}}{4\pi r^{3}}}}
and hence the field
B c = − μ o ∇ ϕ c = μ o 4 π [ 3 r ^ ( r ^ ⋅ m ) − m r 3 ] {\displaystyle {\textbf {B}}_{c}=-\mu _{o}\nabla \phi _{c}={\frac {\mu _{o}}{4\pi }}{\big [}{\frac {3{\hat {\textbf {r}}}({\hat {\textbf {r}}}\cdot {\textbf {m}})-{\textbf {m}}}{r^{3}}}{\big ]}}
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