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the Lorentz force on a charged particle $F=qv \times B$ can explain Flemings left hand rule (motor rule) and the right hand (dynamo rule)

In the left hand rule, the direction of the current gives the velocity vector v to be cross multiplies with the B vector to five the direction of the force vector

In the right hand rule, the direction that the wire is moved in the dynamo, gives the velocity vector of the otherwise stationary electrons and this cross multiplied with the B vector gives the force vector F on the electrons which gives a current

So the Lorentz equation is sufficient to give the direction in both cases. is that correct?

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Yes the equation $F=qv \times B$ does account for the directions of the three components. Take a look at this image from https://en.wikipedia.org/wiki/Cross_product Cross product
For our formula $F=qv \times B$, $v$ is labelled "a" in the diagram and $B$ is labelled "b" in the diagram. So lets say a points east, b points north and a$\times$b points up. Then you can verify with whatever hand rule you want that a proton traveling east through a magnetic field pointing north will experience an upward force. The same diagram works for an electron (or any negative charge in general). For the electron, we take the exact same cross product, but $q$ will be negative so after taking the cross product we multiply the vector by -1. If you multiply a vector pointing up by -1 the vector will point in the opposite direction (down). This can be seen as the purple arrow pointing down in the diagram. You can verify for yourself that an electron heading east in a magnetic field pointing north will experience a downward force.

However if you are using this to help you remember, it may not help as there is no obvious reason why the cross product makes a vector in this direction. In fact the cross product has its own hand rule to explain which way the result vector points.

Personally I find the normal hand rules quite confusing because I can never remember which force goes on which finger etc. My high school physics teacher taught me a trick that I like to use. The first part is to decide that the right hand is positive and the left is negative. This is easy to remember because most people prefer using their right hand (positive experience) over using their left hand (negative experience). The second part is to imagine you are pushing on a door with the palm of your hand flat on the door.

Like in this image free image from http://www.freeimages.com/

Now the three components are easy to remember as follows. The force is obviously in the direction of the force you are applying. Magnetic fields are always represented with multiple lines so the direction of the magnetic field is your 4 fingers. Finally the last one (the velocity) is represented with your thumb.

If you twist your hand so your palm faces up, you will see it agrees with the first image. Then you can use your left hand for an electron.

As for the "motor rule" and the "dynamo rule" I assume you mean the force a wire carrying current feels when in a magnetic field and the current induced in a wire when moved through a magnetic field. These can both be found with the Lorentz force thinking about what happens to a single electron, and then realizing that all the electrons experience force is the same direction so you can just add it up.

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Electromagnetic induction indeed has three possibilities of interaction between two of the three constituents (when non-parallel to each over) to get the third constituent: - a moving charge in a magnetic field induce a sideway movement (deflection) of the charge (Lorentz force) - an electric charge, accelerated in a coil (deflected sideways) induce a magnetic field (generation of a magnetic field) - a wire, moved in a magnetic field, induce an electric current (electric generator).

The equation of Lorentz force is $q\vec{v}\times\vec{B} = \vec{F}$.

According to the rules of vector products, the formula is changeable for orthogonal vectors to the forms

$$(\vec B\times \vec F) / || \vec B|| ^2 = q_e\vec v_e$$

(induction of a current in an electric generator) and

$$(\vec F \times q_e \vec v_e ) / || q_e \vec v_e|| ^2 = \vec B$$

(induction of a magnetic field by the movement of a conductor transversely to the currents direction).

Using this equations one always can use the left hand rule for all three equations. The thumb is the first factor, the index finger is the second factor and the middle is the result. This holds for the flow of the elctrons, for the technical direction of the current (from plus to minus) one has to use the left hand rule.

https://upload.wikimedia.org/wikipedia/commons/a/ac/UVWREGEL_new.png

(not availible in the English Wikipedia, so this is a sketch from the German Wikipedia)

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  • $\begingroup$ It has to be discussed, what $\vec B$ is really. If it is wrong in the last two equations it has to be incompletely in the first equation too. $\endgroup$ – HolgerFiedler Feb 7 '16 at 20:44

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