References: Griffiths, David J. (2007), Introduction to Electrodynamics, 3rd Edition; Pearson Education – Chapter 11, Post 23.
Because Earth’s magnetic north pole is not at the same place as the orbital north pole, a component of Earth’s magnetic dipole rotates as the planet rotates each day. This gives rise to radiation being emitted. Should we worry that Earth is losing a lot of energy in this way?
Suppose the angle (latitude difference) between true north and magnetic north is and the magnetic dipole moment is . Then the component of that rotates has magnitude so we can write this component as
The power radiated by a time-varying magnetic dipole is
where is the magnitude of the magnetic dipole moment. In our case
so the power radiated is
The magnetic field due to a dipole moment is
Taking Earth’s field strength at the equator to be around and taking at the equator and the radius of Earth to be , we get an estimate of Earth’s magnetic dipole moment:
Using and we get
We needn’t worry about Earth losing a significant amount of energy through radiation from its magnetic field.
For a pulsar, however, it’s a different story. A typical pulsar has a radius of around , a magnetic field strength of and a rotation period of giving . This gives
Taking an average value of for we get a power of around . By comparison, the Sun produces ‘only’ around so a pulsar produces the output of 5 billion suns from its magnetic dipole radiation alone.
Not all pulsars generate this much power, though. The period of the crab nebula pulsar is around 33 ms, so (assuming the radius and magnetic field are the same as above) this gives a power of around which is still pretty impressive.