Expert: Daniel Mazur - 1/17/2010

Question
Hi Daniel,

What I know of the earth's magnetism is that it is basically caused by currents inside the surface of the earth caused due to the flow of certain charged compounds.

These currents of course do not have an absolutely steady value and also keep on changing their direction over the years but if we neglect that for a moment and chose a very small time, the currents have a particular mean value from the earth frame which is rotating with respect to its axis.

So they produce a magnetic field of certain value. But if we observe the earth from a frame up in space which "is at rest w.r.t to the axis of the earth" , we wouldn't get the same value of these currents as we would get standing somewhere on the earth.

And so, the value of the magnetic field created by these currents should also have a different value in such a frame...
infact the value should be different for any frame up there which is not rotating with the angular velocity equal to that of the earth w.r.t the axis of the earth.....

Am I correct??

If I am....then how much of a difference is there....is there any considerable difference for frames that are not rotating with the angualr velocity equal to that of the earth w.r.t the axis of the earth...for eg. some artificial satellite???
Are the differences small enough to not cause problems in the workings of such satellites..for eg. communication satellites?? or are these satellites designed taking this effect into account??

on the same lines, do they then have any considerable effect on astronoauts working up there...provided, of course, that they are not rotating with the same angular velocity as that of the earth....??


Thanks
Shikhin


Answer
Hi Shikhin,

the Earth's magnetic field is a sum of field produced by internal currents (especially the molten metallic core) and the atmospheric currents, which exist in the ionosphere and magnetosphere. The latter contribution is by no means not negligible and it has its practical effects and uses. It does affect the optimal height, in which man-made satellites cruise, and the communication channels between the satellites and space shuttles and Earth. They must be and are designed to take the magnetic field and its changes into account. The geostationary satellites feel almost the same magnetic field as is felt on the surface, but not the same - the magnetic field is not homogeneous, so a radial difference of 50-100 km matters. Anyway, there is a huge amount of correct information at Wikipedia, http://en.wikipedia.org/wiki/Earth%27s_magnetic_field ...

Space shuttles and satellites that orbit Earth with other than geostationary velocities perceive a different magnetic field. Magnetism of current carrying conductors is a relativistic effect and, therefore, the state of motion of the observer with respect to Earth or an inertial plane (e.g. one that includes Earth's rotational axis) influences the measured values. How large is the difference? To know exactly you will need to perform some relativistic calculation. The result will, naturally, depend on the magnitude AND direction of the observer's velocity, so without doing the maths I cannot tell. The magnetic field of a dipole is given by this B(m,r) vector equation here http://en.wikipedia.org/wiki/Dipole#Vector_form . Earth's magnetic dipole moment is m_Earth = 8*10^22 Am^2. To derive the magnetic field of this dipole in different systems of coordinates you need to apply the Lorentz transformation, http://en.wikipedia.org/wiki/Lorentz_transformation . As the magnetic axis is not aligned with the rotational axis, the calculation will get lengthy. However, you can plug this into Mathematica, because with the information here (or on Wiki) there is nothing else to do than to perform the symbolic manipulation. As it is such a time-consuming mechanical thing, I will not do it for you, forgive me.

Take care!
Daniel