The rotation of the Earth's dipolar magnetic field produces an electric field in space. Because the electric field is zero in the rotating frame, it is equal to -(wxr)xB in a fixed frame, where w is the angular velocity of the Earth, r the radial distance and B the magnetic field. This formulation given by Hannes Alfven in in 1950 is valid in vacuum. In the case of the Earth, E is the corotation electric field, radially directed to the Earth. A discussion of the formulation can be found in "The External Electric Field of a Rotating Magnet", by G. Backus, Astrophysical Journal, vol 23, p508, 1956.,

The rotation of the Earth's dipolar magnetic field produces an electric field in space.

Because the electric field is zero in the rotating frame, it is equal to $$ \mathbf E=-(\omega\times \mathbf r)\times \mathbf B $$ in a fixed frame, where $\omega$ is the angular velocity of the Earth, $\mathbf r$ the radial distance and $\mathbf B$ the magnetic field. This formulation, given by Hannes Alfven in 1950, is valid in vacuum. In the case of the Earth, $\mathbf E$ is the corotation electric field, radially directed to the Earth. A discussion of the formulation can be found in "The External Electric Field of a Rotating Magnet", by G. Backus, Astrophysical Journal, vol 23, p508, 1956. (ADS link)