I performed an additional calculation that might be of interest to show consistency between equations (18) and (29): For a path along the surface of the sphere, (18) shows that the electric field tangent to the surface will be along the lines of longitude. Thereby, I computed ## E_{\theta}=-\frac{M \omega R}{3 c} \, \sin(2 \theta) ## from (18). (Note: ## -\frac{dU_{r=R}}{R \, d \theta}=E_{\theta} ##). ## \\ ## I then computed the component of ## E_{\theta} ## from (29) for the case of a point on the surface ## r=R ## with ## y =0 ## : The components of ## E_x ## and ## E_z ## along ## \hat{a}_{\theta} ## were computed and added together, and were consistent with ## E_{\theta} ## computed from (18). ## \\ ## (29) applies to ## r \geq R ## while (18) applies to ## r \leq R ##, so that this particular calculation only works for ## r=R ##. ## \\ ## For any EMF measurement, this path on the surface (along the lines of longitude) is really the most important in any case, because the probes need to make contact to the sphere. ## \\ ## Editing: One puzzle that arises from this result: perhaps
@vanhees71 can provide some insight, is in equation (16), there is no z component of the electric field inside the sphere. This radially inward electric field does have a component along ## \hat{a}_{\theta} ##, (as computed from ## E_{\theta}=- \frac{dU_{r=R}}{R \, d \theta} ## ), and thereby the electric field from equation (29), that does have a non-zero z-component for ## r \geq R ##, and a completely equivalent ## E_{\theta} ## at ## r=R ##, appears to be consistent. The electric field ## E ## is not required to be continuous across the boundary because of the surface charge on the sphere. The parallel components of ## E ## do need to be continuous across the boundary though, and they are, as the calculation linking (18) to (29) shows. Inside the sphere we can have ##E_z=0 ## and outside ##E_z \neq 0 ##. ## \\ ## Additional comment: Very interesting calculations by
@vanhees71 computing the EMF for the homopolar generator for a spherical shape. The computations would perhaps be simpler for a cylindrical shape where (18) would be ##U=\frac{M \omega }{2c}(x^2+y^2) ## and the EMF would be across the top face of the cylinder=perhaps not exact because of fringe effects. [Editing: This is rather inexact in the case of a cylinder, because at the endface, the ## B ## field is reduced by a factor of 2 from what it is in the middle]. One additional question I have is, what happens for a thin magnetized disc? The ## B ## field inside the disc is greatly reduced from the ## B ## of a long cylindrical shape, so presumably the EMF that gets generated would be much less than that which is generated from a long cylindrical shape. ## \\ ## Editing: The calculation to compute the surface charge distribution is quite interesting and non-trivial. The Legendre polynomials/multipole expansion which are solutions of the Poisson equation appear to provide the necessary structure to make for the solution. (If I understand it correctly, the surface charge distribution will cause a potential both inside and outside which must consist of multipole expansions, which are the general solution of Poisson's equation in the vacuum). I'm still working through it, but I'm finding it very interesting. Thank you
@vanhees71 :)
