(a) In a source-free region the magnetic scalar potential of a single degree- ,…

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

Problem

(a) In a source-free region the magnetic scalar potential of a single degree-

n

, order-

m

harmonic can contain both an internal part, varying as

(a/r)^{n+1}

with coefficient

g_n^m

(or

h_n^m

), and an external part, varying as

(r/a)^n

with coefficient

q_n^m

. Write

V

as the sum of these two terms and use

\mathbf{B}=-\nabla V

to obtain the radial and horizontal (colatitude) field components at

r=a

in terms of

g_n^m

and

q_n^m

. Show that measuring both a horizontal and the radial component at the surface is enough, in principle, to solve for

g_n^m

and

q_n^m

separately, and hence to decide whether a given harmonic is internal, external, or a mixture. (b) Starting from Ampère's law with the displacement current,

\nabla\times\mathbf{B} = \mu_0\mathbf{J} + \mu_0\varepsilon_0\partial\mathbf{E}/\partial t

, and Ohm's law

\mathbf{J}=\sigma_e\mathbf{E}

for a stationary conductor, form the ratio of the displacement-current term to the conduction-current term for a field varying on a timescale

\tau

. Evaluate this ratio numerically for the liquid outer core, taking

\sigma_e = 10^{6}\,\mathrm{S\,m^{-1}}

\varepsilon_0 = 8.85\times10^{-12}\,\mathrm{F\,m^{-1}}

and

\tau = 1\,\mathrm{yr}

(the timescale of secular variation), and state whether the displacement current may be dropped.

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