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Generally the hand rules are 3D visualizations for vectors, i.e., the direction of magnetic field/flux, the direction of current/+ve charge flow, the direction of resultant force/movement. It's probably too hard for most students to visualize the directions in the head, so we use the hand rules to help figure out the relative directions.

The directions of various vectors are (in O-Levels, anyway) always at right angles to one another.

For Fleming's rules:

The *TH*umb always points in the direction of *TH*rust (or more accurately force)
The *F*irst finger always point in the direction of *F*lux/*F*ield (magnetic field lines)
The se*C*ond finger always point in the direction of *C*urrent (+ve charge flow; or more accurately, EMF)

Which hand we use is determined by the situation -- and there are only two types of situations, i.e. either a MOTOR type situation or a GENERATOR type situation.

* MOTors produce MOTion --> use the LEFT hand rule
* geneRatoRs produce cuRRent --> use the RIGHT hand rule

Observe the lengths of the words. The word LEFT is shorter than the word RIGHT. The word MOTOR is shorter than the word GENERATOR. The hand we use matches the situation. LEFT for MOTOR; RIGHT for GENERATOR. Alternatively, observe the letter R in the words -- geneRatoR produces cuRRent, we use the Right hand rule.

There is another hand rule in use -- the right hand slap/palm rule. This rule is rarely mentioned in any O-level syllabus, but it is the simplest. For the right hand slap rule, place your palm flat on the surface of a table. Your fingers should be parallel to one another. Your thumb should be extended at a right angle to the fingers, Lift up your palm. What you are looking at now is the right hand slap/palm rule. The *F*ingers represents the direction of *F*ield; the thumb represents the direction of input (current/force); the palm faces in the direction of output (force/EMF) The right hand slap/palm rule works for both motors and generators.

The Right Hand Grip/Curl Rule is used for the relationship between current and magnetic field.

For STRAIGHT WIRE, when you grip the wire with the thumb pointing in the direction of current, the fingers point in the direction of induced magnetic field. One thumb - one direction of current; multiple fingers - multiple field lines.

If you grip a SOLENOID/COIL, the grip points in the direction of current, the thumb points at the North end of magnetic field. Again, one thumb - one north; multiple fingers, multiple coils.
 
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