Paschen's Law and E-field between spheres

[Mr Boom]

I am reading up on Paschen's Law and came across some papers measuring breakdown voltage between two identical spherical electrodes. I've read in a couple places that the electric field is uniform when the gap distance is no larger than the radii. Why is that? It seem to me that the field strength would never be uniform.

 

[Jano L.]

You are right, it is not exactly uniform. However, when the distance between the surfaces of the spheres is much smaller than the distance of their centres, the positive and negative charge are concentrated close to each other on the facing surfaces. Now if we take only this part of space, smaller than the radius of the sphere, the surfaces are almost parallel and the situation is similar to plane parallel condenser, in which the electric field would be uniform.