What the force depends on
How big is the force on the wire? The size of the force depends on three factors.
The current
The size of the current flowing in the conductor.
The length
The length of the conductor placed in the magnetic field.
The field
The strength of the magnetic field.
When these three factors increase, the force increases; when they decrease, the force decreases. That is, the force is directly proportional to each of the three.
Why each factor matters
A current is a flow of charges, and the magnetic field pushes on each moving charge. Double the current and twice as many charges pass each second, so the push doubles. Double the length of wire inside the field and twice as many charges are being pushed at once. Make the field twice as strong and each charge is pushed twice as hard. Engineers use all three when they design a motor: they wind the coil with many turns so a great length of wire sits in the field, use strong permanent magnets, and supply enough current. In activity 13.3, a stronger magnet or a second cell would make the wire BC roll faster.
The force needs the wire across the field
The rule of proportion holds with the wire at right angles to the field, as in the activity. A wire lying along the field lines feels no force at all, however big the current.
Current, length, field
The force on a conductor in a magnetic field is directly proportional to the current, the length in the field, and the strength of the field.