The degree-1

PHYSICS · Multi-part · Fermi University Physics · Electricity, Magnetism & Electromagnetism · Earth, Climate & Energy Physics · The solid Earth: structure, fields and thermal history

The Solid Earth, Ch The geomagnetic field: potential, spectrum, induction and the palaeomagnetic record

Problem

The degree-1, zonal (m=0m=0) part of the geomagnetic potential, allowing for both an internal and an external source, is V=a[(ar)2g10+ra q10]cos⁡θ,V = a\left[\left(\frac{a}{r}\right)^{2} g_1^0 + \frac{r}{a}\,q_1^0\right]\cos\theta, where g10g_1^0 is the internal Gauss coefficient, q10q_1^0 the external one, aa is the Earth's radius and θ\theta is colatitude. Recall Br=−∂V/∂rB_r = -\partial V/\partial r and Bθ=−(1/r) ∂V/∂θB_\theta = -(1/r)\,\partial V/\partial \theta. ((a)) Show that at the surface r=ar=a the amplitude (coefficient of cos⁡θ\cos\theta or sin⁡θ\sin\theta as appropriate) of BrB_r is 2g10−q102g_1^0 - q_1^0 and the amplitude of BθB_\theta is g10+q10g_1^0 + q_1^0. ((b)) A global survey of the degree-1 field finds a horizontal-field amplitude corresponding to g10+q10=29 800 nTg_1^0+q_1^0 = 29\,800\,\mathrm{nT} and a radial-field amplitude corresponding to 2g10−q10=59 300 nT2g_1^0 - q_1^0 = 59\,300\,\mathrm{nT}. Find g10g_1^0 and q10q_1^0, and state what the result shows about the source of the field.

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