Emf induced on an eccentric rotating bar

[Termotanque]

Suppose that a conducting bar is rotating with a speed ω, around an axis a distance a from one end, and b from the other. Suppose further that it is immersed in a uniform magnetic field B parallel to the axis of rotation, and that the measured difference of potential between both ends is V. I want to calculate the appropriate B.

I tried calculating an infinitesimal potential difference dV appearing when an infinitesimal slice of the bar of length dr is rotating, such that the electric and magnetic force cancel: dV = ωrB dr, and then integrating this from -a to +b. This approach appears to be incorrect.

Any hints will be appreciated.

 

[LightHero]

The correct approach is to use the Lorentz Force Law. This states that the force on a charged particle in an electric and magnetic field is equal to the product of the charge, velocity, and the cross product of the electric and magnetic fields. In this case, the Lorentz force between two points on the bar is equal to the product of the charge per unit length (the linear charge density) times the angular velocity, times the cross product of the electric and magnetic fields. By equating this Lorentz force to the force due to the voltage difference between the two points on the bar, you can solve for the magnetic field.