Here's a typical description of Motional EMF (from http://teacher.nsrl.rochester.edu/phy122/Lecture_Notes/Chapter32/chapter32.html, edited for brevity; there are many similar descriptions on various sites):
The figure below shows a conducting rod of length L being moved with a velocity v in a uniform magnetic field B:
The magnetic force acting on a free electron in the rod will be directed upwards. As a result, electrons (the blue dots) will start to accumulate at the top of the rod. The charge distribution of the rod will therefore change, and the top of the rod will have an excess of electrons (negative charge) while the bottom of the rod will have a deficit of electrons (positive charge). This will result in a potential difference between the ends of the rod equal to LvB.
Now looking at that description, I can easily see how a longer rod will result in a greater potential difference -- we've got more electrons for the force to act on, and thus more electrons will tend to accumulate at the top of the rod:
My problem is that I would also expect a wider rod to have a similar effect:
Yet the formula (LvB) says width has nothing to do with it.
So my question is: Why doesn't the width of the rod affect the induced potential difference?
The figure below shows a conducting rod of length L being moved with a velocity v in a uniform magnetic field B:
The magnetic force acting on a free electron in the rod will be directed upwards. As a result, electrons (the blue dots) will start to accumulate at the top of the rod. The charge distribution of the rod will therefore change, and the top of the rod will have an excess of electrons (negative charge) while the bottom of the rod will have a deficit of electrons (positive charge). This will result in a potential difference between the ends of the rod equal to LvB.
Now looking at that description, I can easily see how a longer rod will result in a greater potential difference -- we've got more electrons for the force to act on, and thus more electrons will tend to accumulate at the top of the rod:
My problem is that I would also expect a wider rod to have a similar effect:
Yet the formula (LvB) says width has nothing to do with it.
So my question is: Why doesn't the width of the rod affect the induced potential difference?