In air, a charged soap bubble of radius $R$ breaks into 64 small soap bubbles of equal radius $r$. The ratio of mechanical force per unit area of big soap bubble to that of a small bubble is

When a soap bubble of radius $R$ breaks into 64 small bubbles of equal radius $r$, the volume of soap solution remains constant.

$\frac{4}{3}\pi R^3 = 64 \times \frac{4}{3}\pi r^3$
Solving gives $R^3 = 64r^3$ and $R = 4r$.

The mechanical force per unit area inside a soap bubble is the excess pressure due to surface tension, given by $P = \frac{4T}{\text{radius}}$.

For the large bubble, $P_{\text{big}} = \frac{4T}{R}$
For a small bubble, $P_{\text{small}} = \frac{4T}{r}$

The ratio is $\frac{P_{\text{big}}}{P_{\text{small}}} = \frac{r}{R}$
Substituting $R = 4r$ gives $\frac{P_{\text{big}}}{P_{\text{small}}} = \frac{1}{4}$.

Since the problem specifies mechanical force per unit area, electrostatic effects from any charge are disregarded despite the mention of a charged bubble as they require additional parameters not provided.

The ratio is $1:4$.