(1)The magnetic field around a conductor where there's a current can easily be demonstrated using something like a compass needle. The magnetic field lines will be perpendicular to the direction of the current.
Now, charges will not be affected by a electromagnetic force unless they are moving. However, in my textbook they say when the velocity is perpendicular to some magnetic field they are in, a force will be perpendicular to the charges - direction given by right-hand rule.
Obviously that can as well be demonstrated very easily through experiments. But is there a way to show this theoretically with magnetic field lines, i.e. why the force will be directed upwards in some cases for example?