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I get it. The drift speed of electrons is V=J/n.e where J is the current density and n is the number of charges per unit volume.
If I is the current in a conductor of cross-section s, then J = I/s, hence V = I/s.n.e
If d is the thickness of the sample and w its width, then (1) V = I/d.w.n.e
The electric field is E=VxB, i.e. E=V.B when V and B are orthogonal.
The Hall voltage is U = E/w = V.B/w, hence (2) V=U.w/B
Since (1) = (2) => I/d.w.n.e = U.w/B => I/d.n.e = U/B => U = B.I/d.n.e
So we can see that U is inversely proportional to thickness d.
And all this was in fact trivial and obvious from the start. In fact, the number of charges involved is smaller when the volume containing them is smaller. So for the same current I, i.e. for the same number of charges passing per unit of time, if the conductor cross-section is smaller, it means they're moving faster!
For example, a current I of N charges passing at speed V is strictly equivalent to a current I of N/2 charges passing at speed 2.V .
Reducing the cross-sectional area of a conductor for the same current increases the drift speed of the electrons, and therefore the effect of the Lorentz force on each electron. This has no effect on the overall Lorentz force when translated into a mechanical force, since this force on a smaller number of electrons flowing faster is equivalent to that on a larger number of electrons flowing slower. But contrary to our intuition, it does have an effect on the Hall voltage.
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"Open your mind, but not like a trash bin"
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Author
Topic: The Inventions of Willi von Unruh and Hans Coler (Read 14736 times)
