A capacitor has square plates, each side a, making a small angle theta with each other. Determine the capacitance C in terms of a, d, and theta. The capacitor may be divided into differential strips which are effectively in parallel.

To determine the capacitance C in terms of a, d, and theta, we can follow these steps:

1. Start by dividing the capacitor into differential strips, which are effectively in parallel. This means we can calculate the capacitance of a single strip and then integrate over all the strips to find the total capacitance.

2. Consider a single differential strip within the capacitor. The differential strip will have a length dx and a width dy (infinitesimally small). The distance between the plates at the location of the strip is d + y * tan(theta), where d is the distance between the plates at the center.

3. The differential strip can be visualized as a rectangle with dimensions dx and dy. The area of this strip is A = dx * dy.

4. To calculate the capacitance of the strip, we need to determine the electric field between the plates. For this configuration, the electric field is not constant. However, we can approximate it as constant over the length of the strip. Therefore, the electric field between the plates is E = V / (d + y * tan(theta)), where V is the voltage applied across the plates.

5. The electric field E is related to the charge on the plates Q and the area of the strip A by the equation E = Q / (ε0 * A), where ε0 is the permittivity of free space. Rearranging this equation, we have Q = E * ε0 * A.

6. The capacitance of the strip is defined as C = Q / V. Substituting the expression for Q from step 5, we get C = (E * ε0 * A) / V.

7. Substituting the expressions for E and A from steps 4 and 3, we have C = (V / (d + y * tan(theta))) * ε0 * (dx * dy).

8. Integrating over all the strips, we obtain the total capacitance C of the capacitor. The limits of integration for y would be from -a/2 to a/2, and the limits of integration for x would be from 0 to the width of the plates.

9. The total capacitance C is given by the double integral of the expression obtained in step 7, with the appropriate limits of integration.

It is important to note that depending on the specific values of a, d, and theta, the integral may not have a closed-form solution and may require numerical methods for evaluation.