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Let's say we have a parallel plate capacitor, and a solid dielectric that partially fills the space between the plates.

If the dielectric is just a slab of material, with sides parallel to the plates, then I can calculate the capacitance easily. What happens if instead it is slightly tilted? Does the capacitance stay the same? And what if instead of a slab of material, the dielectric has a completely irregular shape, but occupies the same volume as before? Is the capacitance different?

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In the tilted/irregular shaped dielectric capacitor, your dielectric becomes a combination of the dielectric material and air/vacuum. You can then calculate an effective permittivity of the medium.

In general, this will be lower than the uniform block of dielectric since air will have a lower $\varepsilon$ than the dielectric, and therefore a lower capacitance. Under certain assumptions, you can estimate the effective dielectric constant using effective medium approximations, such as Maxwell-Garnett. To be most accurate, or in cases where the air gaps are large or geometry complex, you really need to put it in a finite element model and solve it computationally.

Note too that the capacitance of a parallel plate capacitor is not really "easy" to calculate... the textbook problem ignores fringing fields for which analytic solutions are known only for the simplest geometries. Again, numerical simulations are needed.

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