6.5.24∗. A uniformly charged sheet having the shape of an isosceles right triangle was folded in half. In this case, work $A$ was performed against the forces of the electric field. What kind of work do you need to do to fold the resulting triangle again in the same way?
Solution
By folding an isosceles right triangle in half, its new linear dimension is $1/\sqrt2$ of its original dimension. If the electrical potential energy of the original charged triangle is $U$, then the potential energy of the new triangle is $\sqrt2\,U$, because potential energy varies inversely with distance between charges while total charges remain the same. Thus, the work done in the first folding is $A=\sqrt2\,U-U$. If the new triangle is folded once more, then the potential energy of the last triangle is $(\sqrt2)^2\,U$, and the work done in the second folding is $(\sqrt2)^2\,U-\sqrt2\,U=\sqrt2A$.