The basis of MHD analysis is Maxwell's equations. Faraday's law of inductance relates the electric (E) field to a time-varying magnetic (B) field:
Ampère's law describes how currents (J) and time-varying electric fields produce a magnetic field:
where ^0 is the magnetic permeability of free space (=4p X10-7 N A-2) and e0 is the permittivity of free space (= 8.85X10-12 C2 N-1 m-2). In most planetary magnetic fields, i0e0 (dE/dt) is very small and can be ignored. The current (J) induced by the magnetic field is given by Ohm's law:
where r is the electrical conductivity of the material and v is the velocity of the conducting fluid. Combining Eqs. (3.32), (3.33), and (3.34) gives the induction equation:
Equation (3.35) shows how the magnetic field B varies with time. The first term on the right shows that the conducting fluid must be moving with velocity v in order to maintain the magnetic field. The second term on the right is called the diffusion term and 1 / (ior) is the magnetic diffusivity. In the absence of fluid motions (i.e., v=0), the magnetic field decays over some time period related to the magnetic diffusivity. For terrestrial planets, this decay period is of order 104 years. Since this is much less than the ~4.5 Ga age of the Solar System, planets with active magnetic fields such as Earth must be generating their fields at the present time.
During aerobraking, MGS's Magnetometer/Electron Reflectometer (MAG/ER) experiment became the first instrument to obtain magnetic field observations below the planet's ionosphere. The maximum magnitude of Mars' dipole is ~2 X1017 A m , corresponding to a magnetic field strength of 0.5 nT at the equator (Acuña et al., 2001). This small value indicates that Mars does not currently have an active magnetic dynamo.
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