How would Earth's magnetic field change if the outer core was solid?
We have an excellent reference of what would happen to the Earth's magnetic field if it had a solid core: we can look at Venus' magnetosphere. Its internal magnetic field is dominated by the remanent crustal magnetization and its external magnetic field by the induced currents and their magnetic field in the ionosphere. The magnetic field mostly misses any dynamo part due to Venus' very slow rotation (even though it also has a similar internal structure like Earth).
Thus as a result, the Earth's magnetosphere would shrink considerably had it as little rotation and/or no liquid outer core. With only a solid core remaining, it would also only be able to produce some small remanent magnetization which is usually MUCH less than any self-feedback dynamo like Earth, Jupiter, Sun,... are able to produce.
Erosion of the upper atmosphere would be much stronger and we could enjoy polar lights about continuously and even at low lattitudes as we'd loose considerable magnetic field strength, especially of the dipole moment.