I'm trying to get a better intuitive understand of this formula:

$$e = (\overrightarrow {u}\times \overrightarrow {B}) \cdot \overrightarrow {l} $$

$(\overrightarrow {u}\times \overrightarrow {B})$ will give a vector that's perpendicular to the two. Do I use the right-hand rule, where the thumb is in the direction of the velocity and the four fingers are in the direction of the magnetic field? Give me a perpendicular vector out of my palm?


$e$ is the induced voltage, $l$ is the length of the conductor, $u$ is the velocity, and $B$ is the magnetic flux density.

This formula is also known as "generator action" (electromechanical circuits)

  • $\begingroup$ I can't follow your description of the right hand rule. Try this: extend your forefinger and thumb to make a right angle, then stick out your middle finger perpendicular to the two others. Point fore-finger along $\vec{u}$, middle finger along $\vec{B}$. Then the thumb points along $\vec{u}\times \vec{B}$ $\endgroup$ – garyp Oct 1 '16 at 20:55
  • $\begingroup$ Lol that's hard. Mine is this: Thumb in the direction of u. And stick your four fingers out and face in the direction of B. Makes a right angle between the thumb and four fingers. Then the force would be coming out of your palm. $\endgroup$ – user367640 Oct 1 '16 at 21:07
  • $\begingroup$ Lol. Huh? :) Easy is in the eye of the beholder, I guess. $\endgroup$ – garyp Oct 1 '16 at 23:14

This formula derives from the Lorentz force formula. $\vec{u}\times \vec{B}$ is a cross product of the vectors which is also a vector. You get the direction of this vector by putting both vectors at the same origin and turning the tip of the first to the direction of the second vector. The direction of a right winding screw is moved is the direction of the product vector. The induced emf is proportional to lenghth l.

  • $\begingroup$ I thought the lorentz was this: $f = (\overrightarrow {u}\times \overrightarrow {B}) q $, I understand how the lorentz force works. But for the generator action I dont understand the dot product of the length of the conductor. $\endgroup$ – user367640 Oct 1 '16 at 20:58
  • $\begingroup$ The dot product with the length vector means that only the Lorentz force component in the direction of the length of the wire (projection) contributes to the emf e. $\endgroup$ – freecharly Oct 1 '16 at 21:08

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