Capacity vs measure of a set - intuitive understanding

Question

There is a concept of measure of "largeness" of a set, called capacity. The intuition is, instead of physical largeness (measured by Hausdorff or Lebesgue measure), capacity measures how good a given set is in at holding charge. Analytically, the $2$-capacity of a set $\Omega$ sitting inside a Riemannian manifold $M$ is given by $$cap_2(\Omega) = \inf_{u \in C^\infty_0(M), u|_\Omega \equiv 1} \int_M |\nabla u|^2 dM.$$ In the special case $M = \mathbb{R}^3$, I would like to intuitively understand what large capacity actually means. By the heuristic of ability to contain charge, it seems to me that a disc on the surface of the sphere $S^2$ would have larger capacity than that same (isometric) disc if it were rolled and folded upon itself. As another example, consider a book held open so that the angle between the pages is $180^\circ$. Intuitively it seems that the open surface of the pages now have a higher capacity than if the book were partially closed to make the angle, say, $45^\circ$. Is this intuitive understanding correct? How do I prove these rigorously from the definition?