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Fleming's left hand rule (for electric motors) shows the direction of the thrust on a conductor carrying a current in a magnetic field.
The left hand is held with the thumb, index finger and middle finger mutually at right angles. It can be recalled by remembering that "motors drive on the left, in Britain anyway."
The First finger represents the direction of the Field.
The Second finger represents the direction of the Current [conventional current, positive(+) to negative(-).
The Thumb represents the direction of the Thrust or resultant Motion.
This can also be remembered using "FBI" and moving from thumb to second finger.
The thumb is the force F
The first finger is the magnetic field B
The second finger is the of current I
There also exists Fleming's right hand rule (for generators). The appropriately-handed rule can be recalled by remembering that the letter "g" is in "right" and "generator".
Both mnemonics are named after British engineer John Ambrose Fleming who invented them.
Other mnemonics also exist that use a left hand rule or a right hand rule for predicting resulting motion from a pre-existing current and field.
There is an excellent mnemonic, pioneered many years ago in Canada, which does away with the need for either of Fleming's complex rules. Imagine that electrons are eskimos, running through a forest. The fir trees look like arrows pointing up, and represent lines of force (ie a north pole is beneath the eskimos). As the eskimos run they always veer towards the left. That is the only rule needed.
Hold a north pole at the left side of a cathode ray tube tv, in which electrons are rushing toward the screen. The lines of force run, roughly, from left to right across the tube. The electrons will turn to their left in this "forest", ie downwards, and therefore the image will move down. A south pole will make the image rise.