We're going to die.
Surface of the Frozen Earth
This picture depicts a frozen over Earth that will eventually occur.
This is a futuristic view of what the Moon may look like from the Earth if it freezes over.
Once the Earth freezes over, blackouts often occur. No Magnetic field means no protection against space radiation. We will have more cancers and more blackouts. But that's not the the big issue. The worst part is our atmosphere will begin to vanish.
The Earth Goes Cold
Once the core has solidifed it can no longer generate a magnetic field. There may be some frozen in field, but I would guess that the strength of the magnetic field with decrease as more and more of the core freezes, so any residual field is likely to be small. The core of Mars is thought to be frozen and it's magnetic field is negligable.
Anyhow, the main problem would be that without the magnetosphere to protect it the atmosphere will gradually be stripped by the solar wind and we'll be left with nothing to breathe. In addition, without an atmosphere the surface will be subject to wide temperature swings; just as the Moon is in fact.
Very gradually mountains will shrink. Grain by grain, they will crumble down and get washed away by rain and rivers into the sea.
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