  Magnetic fields, which are undoubtedly present in extragalactic jets and
responsible for the observed synchrotron radiation, can affect the morphology
and dynamics of the jets and their interaction with the ambient cluster medium.
We examine the jet propagation, morphology and magnetic field structure for a
wide range of density contrasts, using a globally consistent setup for both the
jet interaction and the magnetic field. The MHD code NIRVANA is used to evolve
the simulation, using the constrained-transport method. The density contrasts
are varied between \eta = 10^{-1} and 10^{-4} with constant sonic Mach number
6. The jets are supermagnetosonic and simulated bipolarly due to the low jet
densities and their strong backflows. The helical magnetic field is largely
confined to the jet, leaving the ambient medium nonmagnetic. We find magnetic
fields with plasma \beta \sim 10 already stabilize and widen the jet head.
Furthermore they are efficiently amplified by a shearing mechanism in the jet
head and are strong enough to damp Kelvin-Helmholtz instabilities of the
contact discontinuity. The cocoon magnetic fields are found to be stronger than
expected from simple flux conservation and capable to produce smoother lobes,
as found observationally. The bow shocks and jet lengths evolve self-similarly.
The radio cocoon aspect ratios are generally higher for heavier jets and grow
only slowly (roughly self-similar) while overpressured, but much faster when
they approach pressure balance with the ambient medium. In this regime,
self-similar models can no longer be applied. Bow shocks are found to be of low
excentricity for very light jets and have low Mach numbers. Cocoon turbulence
and a dissolving bow shock create and excite waves and ripples in the ambient
gas. Thermalization is found to be very efficient for low jet densities.
