This is well known to everyone that clouds carry electric charges and cumulonimbus clouds contain huge amount of electricity. It is also known from Maxwell's equations that moving particles induce magnetic field around them. I am sorry if I am missing any concept here but I have a question,
Should there also be a magnetic field around cumulonimbus clouds due
to the presence and motion of charged particles in them?
I want to know about field studies reporting the intensity of the magnetic field in different cloud systems. I am interested in knowing about cold clouds as the interaction of solid hydrometeors could be prominent in them. I also want to know if the intensity is sufficient enough to have an interaction with magnetic field of earth. I see recent modeling studies are being performed on electric fields, but the mention of magnetic field is still missing in them.
The short answer to your question is yes all charge particles (positive or negative), independent (like electron, proton) or attached to other materials like cloud droplets, molecules, atoms, etc. when in motion produce magnetic field.
If two charged particles of same charge are moving in opposite direction then they cancel out each other's magnetic field. Hence, the random motion of droplets in cloud cause net zero magnetic field away from the cloud. One can understand this by analogy to a pair of electric wires used in powering homes which does not produce appreciable magnetic field at distance when the two wires of pair are close to each other because electrons are moving in opposite directions in the two wires.
In organized motion of charged particles that is when there is net movement of charge in a preferential direction, for example a cloud with charged droplets moving under influence of wind one can have net magnetic field. However, here too, if the cloud as whole contains both positive and negative charges in equal numbers will not produce appreciable magnetic field at a long distance because magnetic field produced by negative charges will be opposite in direction with respect to magnetic field produced by positive charge moving in same direction. If there is net positive or negative charge on cloud then the magnitude of magnetic field will depend on wind speed, which for atmospheric range of wind speeds will produce very small magnetic field.
The appreciable magnitude of magnetic field is produced during lighting since large amount of charge moves along a particular path with high velocity but such movements are momentary and as soon as the lighting strike is over, the magnetic field produced starts dying. If a compass needle is having significant inertia then before the needle can move under influence of newly produced magnetic field, the magnetic field will vanish and hence ordinary compasses are unlikely to show effect of lightning strikes.
Fundamentals of Lightning by Vladimir A. Rakov, Cambridge Univesity Press, ISBN 978-1-107-07223-7, 2016