Fleming's left-hand rule for motors is one of a pair of visual mnemonics, the other being Fleming's right-hand rule (for generators). They were originated by John Ambrose Fleming, in the late 19th century, as a simple way of working out the direction of motion in an electric motor, or the direction of electric current in an electric generator.
When current flows through a wire, and an external magnetic field is applied across that flow, the wire experiences a force perpendicular both to that field and to the direction of the current flow. A left hand can be held, as shown in the illustration, so as to represent three mutually orthogonal axes on the thumb, first finger and middle finger. Each finger is then assigned to a quantity (mechanical force, magnetic field and electric current). The right and left hand are used for generators and motors respectively.
Several memory aids have been used in order to remember the quantity each finger represents.
Van de Graaff's translation of Fleming's rules is the FBI rule, easily remembered because these are the initials of the Federal Bureau of Investigation.
This uses the conventional symbolic parameters of F (for Lorentz force), B (for magnetic flux density) and I (for electric current), and attributing them in that order (FBI) respectively to the thumb, first finger and second finger.
Of course, if the mnemonic is taught (and remembered) with a different arrangement of the parameters to the fingers, it could end up as a mnemonic that also reverses the roles of the two hands (instead of the standard left hand for motors, right hand for generators). These variants are catalogued more fully on the FBI mnemonics page.
Fleming's left-hand rule is used for electric motors, while Fleming's right-hand rule is used for electric generators.
Different hands need to be used for motors and generators because of the differences between cause and effect.