A ring of charge with uniform charge density is completely enclosed in a hollow donut shape. An exact copy of the ring is completely enclosed in a hollow sphere.

A) What is the ratio of the flux out of the donut shape to that out of the sphere?

According to Gauss's theorem, total flux coming out of a closed surface area is equal to 1/ε₀ times charge enclosed

This flux coming out of enclosed charge does not depend upon the shape of enclosing surface area. It only depends upon the charge enclosed.

It the present context, in both the case ie in case of donut or sphere, since amount of charge enclosed is same, flux coming out too will be same in both the case

Hence the required ratio = 1 : 1