Question

A planetary magnetosphere has a dipole magnetic field with a dipole moment M. The dipole axis is aligned along the rotational axis of the planet. All of the plasma in the magnetosphere rotates with the planet at an angular velocity ω. Using spherical (r,θ,ϕ) coordinates derive equations for the following.

(a) The E×B drift velocity vₑ=ω×r.
(b) The electric field E.
(c) The electrostatic potential ϕ.

Answer 1

(a) The E×B drift velocity
`\(v_e = \omega * r\)` in spherical coordinates is given by
`\(v_e = \omega r \sin \theta \hat{\phi}\).`

(b) The electric field E in spherical coordinates is given by
`\(E = -\\abla \phi - (\partial A)/(\partial t)\)`, where
`\(\phi\)` is the electric potential and A is the magnetic vector potential.

(c) The electrostatic potential
`\(\phi\)` in spherical coordinates is given by
`\(\phi = -(M \cos \theta)/(r^2)\).`

Step-by-step explanation:

(a) The E×B drift velocity
`\(v_e\)` is the cross product of the angular velocity
`\(\omega\)` and the position vector r. In spherical coordinates, the position vector r can be represented as
`\(r = r \sin \theta \hat{\phi}\)`, where r is the radial distance,
`\(\theta\)` is the polar angle, and
`\(\hat{\phi}\)` is the azimuthal unit vector. Therefore,
`\(v_e = \omega * r\)` becomes
`\(v_e = \omega r \sin \theta \hat{\phi}\).`

(b) The electric field E can be expressed in terms of the electric potential \(\phi\) and the magnetic vector potential A as
`\(E = -\\abla \phi - (\partial A)/(\partial t)\)`. In spherical coordinates, the electric potential
`\(\phi\)` is given by
`\(\phi = -(M \cos \theta)/(r^2)\)`, where M is the dipole moment. The magnetic vector potential A can be chosen to be zero for a dipole field. Therefore, the electric field E in spherical coordinates is determined by the gradient of
`\(\phi\)` and the time derivative of A.

(c) The electrostatic potential
`\(\phi\)` for a dipole magnetic field in spherical coordinates is given by
`\(\phi = -(M \cos \theta)/(r^2)\)`. This expression represents the electrostatic potential associated with the dipole field. The negative sign indicates that the potential is lower at points where the magnetic field lines exit the planet (southern hemisphere) and higher at points where the field lines enter the planet (northern hemisphere).