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### Dipole field lines

2

1.5

1

0.5

0

−0.5

−1

−1.5

−2 −4

−3

−2

−1

• 0

1

2

3

4

Figure 3: The field lines from a magnetic dipole at the origin.

# 4.3 Planetary magnetic fields

Planets with fluid interiors (Mercury, Earth, Jupiter, Saturn, Uranus, Neptune) generate magnetic fields deep in their cores (why and how these fields are generated is briefly discussed in section 4.9 below). On the planetary surfaces, the fields are fairly dipol-like, since other terms in the multipole expansion decays faster with distance from the core. If space around the planets where a vacuum, we would expect that the fields would become more and more dipole-like with increasing distance from the planet, but this is in general not the case. Due to currents generated in the plasma sur- rounding the planets when it interacts with the solar wind, the magnetic fields are distorted to form magnetospheres.

4.4

# Field transformations

## The relativistic (Lorentz) transformations of the electromagnetic fields between a system S

and

another system S moving with velocity v as seen by an observer in S are

## ET =

1 1 v2/c2

(E + v × B)

(52)

BT

=

1

1 v2/c2

B

c 1 2 v × E

(53)

for the transverse part of the fields, i.e. the components perpendicular to v, and

EL = EL

(54)

BL = BL

(55)

E =E+v×B

(56)

B = B.

(57)

for the longitudinal components (along v). In the non-relativistic limit v/c −→ 0 this reduces to the

## Galilean transformation equations

The Lorentz force q v × B one a charged particle in a magnetic field thus is the force due to the electric field in the reference frame of the particle.

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