Solve for the total charge C = 50 + 20k and the side s of a square with diagonal d = 10 cm.

Let $s$ be the side length of the square and $d$ be the length of its diagonal. A square's diagonal divides it into two right-angled isosceles triangles. By the Pythagorean theorem, the relationship between the side and the diagonal is:

$$s^2 + s^2 = d^2$$ $$2s^2 = d^2$$

We are given that the diagonal length $d = 10 \text{ cm}$.

$$2s^2 = (10 \text{ cm})^2$$ $$2s^2 = 100 \text{ cm}^2$$

Divide both sides by 2:

$$s^2 = \frac{100 \text{ cm}^2}{2}$$ $$s^2 = 50 \text{ cm}^2$$

Take the square root of both sides:

$$s = \sqrt{50 \text{ cm}^2}$$ $$s = \sqrt{25 \times 2} \text{ cm}$$ $$s = 5\sqrt{2} \text{ cm}$$

The side of the square is $\boxed{5\sqrt{2} \text{ cm}}$.