Quick questions on Magnetism and moving charges: WACE Year 12 Physics Unit 3

A charge $q$ moving with speed $v$ at an angle $\theta$ to a magnetic field $B$ experiences a force

When a charge enters a uniform field at right angles, the constant perpendicular force supplies a centripetal force, so the charge follows a circular path. Setting the magnetic force equal to the centripetal force,

A current is simply moving charge, so a straight wire of length $L$ carrying current $I$ in a field $B$ feels

Place a rectangular current loop in a magnetic field and the two sides carrying current across the field feel opposite forces, producing a turning effect (a torque) that rotates the coil. This is the motor effect. A split-ring commutator reverses the current every half turn so the torque keeps driving the coil the same way, giving continuous rotation. The torque is largest when the coil plane is parallel to the field and zero when the plane is perpendicular, which is why a real motor uses many turns and several coils to smooth the output.