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Sol exhibits a magnetic field, most of the planets in orbit around Sol exhibit a magnetic field - strong and weak both.

Does the solar system as a whole exhibit a magnetic field?

Does the paradigm scale upwards to larger celestial systems such as a cluster or galaxy?

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The solar system does posses a magnetic field, although this isn't "intrinsic" in the sense that the planets and the sun have an intrinsic magnetic field.

The sun ejects charged particles called solar wind. Since moving charged particles are essentially a current, they produce a magnetic field. Although as the linked wikipedia article says, the value of this magnetic field at the distance of the earth is tiny (~ $10^{-9}$ Tesla).

Magnetic fields do scale to larger celestial systems like galaxies and clusters of galaxies. They are caused by similar processes. In galaxy cluster there are large masses of mobile plasma that produce a current, which in turn causes a magnetic field to form. It is precisely this magnetic field that causes astrophysical synchrotron radiation that is the emission mechanism that a large part of radio astronomy depends on.

In galaxies the physics is a little more complicated. It is thought that there is a conversion of mechanical energy into magnetic energy. Since we do observe large scale magnetic fields in the Milky Way and other galaxies, there has to be a mechanism to maintain it. The details are quite long to explain in detail here, and seem to me to be covered adequately in that last link, if you want to read more.

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