Two soap bubbles of radius 2 cm and 4 cm , respectively, are in contact with each other. The radius of curvature of the common surface, in cm , is ________ .

We need to find the radius of curvature of the common surface when two soap bubbles of radii 2 cm and 4 cm are in contact.

When two soap bubbles of radii $$r_1$$ and $$r_2$$ (where $$r_1 < r_2$$) are in contact, the radius of curvature of the common surface is given by:

$$\frac{1}{R} = \frac{1}{r_1} - \frac{1}{r_2}$$

This is because the excess pressure inside the smaller bubble is greater, and the common surface bulges into the larger bubble.

$$r_1 = 2$$ cm, $$r_2 = 4$$ cm

$$\frac{1}{R} = \frac{1}{2} - \frac{1}{4} = \frac{2-1}{4} = \frac{1}{4}$$

$$R = 4$$ cm

The radius of curvature of the common surface is 4 cm.