The two square plates of a capacitor, each $10\ \text{cm}$ by $10\ \text{cm}$, are not quite parallel. Along one edge they are $1.0\ \text{mm}$ apart, and the gap grows steadily to $3.0\ \text{mm}$ along the opposite edge, as in the figure. There is air between them. The tilt is small, so treat each narrow strip of the plates, parallel to those edges, as a small parallel-plate capacitor of its own gap. The plates are joined to nothing else, and the effects at the edges can be ignored. Take $\varepsilon_0 = 8.85 \times 10^{-12}\ \text{C}^2\,\text{N}^{-1}\,\text{m}^{-2}$. What is the capacitance?

Fig. 1: Side view of two square plates, each 10 cm long, that are not quite parallel. The gap is 1.0 mm at the left edge and grows steadily to 3.0 mm at the right edge. The tilt is drawn larger than to scale. A narrow strip of width dx at a distance x from the left edge is marked.