Imagine two plane capacitors. One having plates the texture of sandpaper and the other having perfectly flat plates. Assuming similar distance between the plates in both cases will the difference in texture affect capacitance ? Does there exists any experimental data about this. Also too what extent does the roughing up (if it does) affect the capacitance
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2 Answers
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Does roughing up a capacitor aid it's capacitance?
Assuming that the distance between the plates is much greater than the distance between peaks and valleys of the roughened surfaces, it does not.
We can start with the formula for the capacitance, $C=\frac Q V$. If we show that the roughening of the surface does not change the voltage between the plates, we'll prove that the roughening does not change the capacitance.
The voltage between the plates is determined by the field strength and the distance between the plates ($V=Ed$). The field strength is a function of the surface charge density $(E=\frac \sigma {\epsilon_0})$.
Since the surface charge density would not change just because the surface was roughened, we can conclude that the field strength would not change and, therefore, assuming that the roughening did not affect the distance between the plates, the voltage would not change either.
So, since for a given charge level, the voltage between the plates does not change, we can conclude that the capacitance won't be affected.
If the height between the peaks and valleys developed due to the roughening is comparable with the distance between the plates, the capacitance may slightly decrease, because the average distance between the plates will noticeably increase.
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By roughing it up, the area is increased which implies capacitance might increase, but the way the derivation of capacitance works, that's not the case. The area in the derivation is where the E field is crossing inside the dielectric rather than the plates themselves.
Activated carbon is used in some capacitors for it's very large "rugh" surface area, but those operate on a more complicated process than simple plate capacitors. Those are like half capacitor, half battery.
If the dielectric is stronger than the ability of the material of the plates to hold a "surface charge" (max charge per area), then the max energy can be increased by "roughing it up" to give it more surface area, but that kind of material is not likely to be a practical plate. Metals can store much more surface charge than any dielectric can withstand, so it does not help there.
Activated carbon aside, roughing it up does not usually increase the max energy store-able because the field strength at the tips of the roughness concentrates the electric field which will cause the dielectric to break down at a lower voltage.
