# Current induction on a volumic body

The Faraday's induction law states that the temporal variation of the magnetic flux induces an electromotive force, according to the following expression:

$$\varepsilon^i=-\frac{d\theta^m}{dt}$$

I know how this law works on a whorl, coil, but not for a volumic body. How I should write the Faraday's induction law for a solid cylinder? I'm studying induction motors, and so, I need to know how is current inducted on the rotor (cylinder). Should I assume that the current only appears on the surface? Should I assume some "whorl density" like we do for the coil?

I can't find any information on internet about this topic. Do you know some interesting books or sites?

• Are you absolutely sure you know how to apply it for a disk? – Timaeus Feb 27 '16 at 22:31
• No... I thought that I could assume the disk as a sum of independent whorls, and so we have $\varepsilon^i=\varepsilon^i(\rho)$ where $\rho$ is the distance to the center of the disk. But the disk is a conductor, and maybe this thought is wrong. I will remove that section from the question – Élio Pereira Feb 27 '16 at 22:38
• I was trying to find some theory behind the current induction on the rotor of the induction motor, but I didn't have success... – Élio Pereira Feb 27 '16 at 22:44
• You need to know the actual electric and magnetic fields and to find the current flowing through a solid. So you need to learn how to find electric and magnetic fields (and the velocities of the charges) so you can find the EMF in a solid. – Timaeus Feb 27 '16 at 23:02

It doesn't hold for a cylinder, instead you have to compute: $$\oint \left(\vec E+\vec v\times \vec B\right)\cdot \mathrm d\vec \ell.$$
Faraday's Law says $$\oint \vec E\cdot\mathrm d \vec\ell=-\iint \frac{\partial \vec B}{\partial t}\cdot \mathrm d\vec A.$$
And the universal flux rule: $$\oint \left(\vec E+\vec v\times \vec B\right)\cdot \mathrm d\vec \ell=-\frac{\mathrm d}{\mathrm d t}\iint \vec B\cdot \mathrm d\vec A$$ only holds for thin wires (where the charges stay inside and there are no magnetic monopoles), not for solid objects.