- #1
gvjt
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I'll spare you all the details, but suppose for some reason (too complicated to get into here) I needed to design a capacitor with certain fixed parameters and only one adjustable aspect, and I wanted the design to maximize the total charge the device could store. According to my calculations, changing the adjustable parameter described below does not seem to make any difference to the answer, and I'm suspicious that I am making a error, hence my post:
Here are the fixed parameters:
- the plate area A
- the distance between the plates d
- the space between is filled with two dielectrics as follows:
i) Strontium Titanate, (K=300, V/m=8e6)
ii) dry air @ STP, (K=1, V/m=3e6)
- the StrontiumTi is in the middle with equal sized air gaps before each plate
- the air gaps are required in the design and must be at least 0.001d each.
The question becomes, what fraction of "d" should the thickness of the StrontiumTi be to maximize the total charge the capacitor can hold. The issue is that C increases with more StontiumTi, but the air gap has a higher voltage gradient and the maximum charging voltage falls as the air gap decreases. Since Q=CV, I suspect that Q is a constant in this case, but I wanted to set up the equations to prove this, and I'm having some trouble making sense of it all. Now don't ask why the air gap is required, that's an independent issue outside the scope of the problem and is simply a design constraint. If it could be eliminated, then clearly the entire dielectric could just be the StontiumTi and then this would give the ideal situation. And this is why I'm confused, because it seems as soon as we insist on the presence of the air gap, the amount of the other material doesn't seem to affect the total charge attainable. On the one hand, this result seems plausible because the voltage gradient is much lower in the StrontiumTi leaving the bulk of the charging voltage across the thin air gap limiting the charging voltage substantially. But on the other hand it seems fishy that things would change so suddenly and drastically due to the presence of this required air gap, and that's why I'm suspicious that something is amiss. If the air gap wasn't there, Vmax would become 8e6 X d and C would be a maximum giving Q=CV the highest possible value. However, once the air gap is introduced, Vmax drops substantially. As the air gap is increased, Vmax increases, but C falls accordingly, and Q seems to be constant as a result. In my attempts at working on this, I recognized that the voltage gradient (delta V / delta d) is 300 times lower in the StontiumTi than in the air, but that the voltage is the gradient X thickness of the layer. So I think I did all that part correctly.
-gt-
Here are the fixed parameters:
- the plate area A
- the distance between the plates d
- the space between is filled with two dielectrics as follows:
i) Strontium Titanate, (K=300, V/m=8e6)
ii) dry air @ STP, (K=1, V/m=3e6)
- the StrontiumTi is in the middle with equal sized air gaps before each plate
- the air gaps are required in the design and must be at least 0.001d each.
The question becomes, what fraction of "d" should the thickness of the StrontiumTi be to maximize the total charge the capacitor can hold. The issue is that C increases with more StontiumTi, but the air gap has a higher voltage gradient and the maximum charging voltage falls as the air gap decreases. Since Q=CV, I suspect that Q is a constant in this case, but I wanted to set up the equations to prove this, and I'm having some trouble making sense of it all. Now don't ask why the air gap is required, that's an independent issue outside the scope of the problem and is simply a design constraint. If it could be eliminated, then clearly the entire dielectric could just be the StontiumTi and then this would give the ideal situation. And this is why I'm confused, because it seems as soon as we insist on the presence of the air gap, the amount of the other material doesn't seem to affect the total charge attainable. On the one hand, this result seems plausible because the voltage gradient is much lower in the StrontiumTi leaving the bulk of the charging voltage across the thin air gap limiting the charging voltage substantially. But on the other hand it seems fishy that things would change so suddenly and drastically due to the presence of this required air gap, and that's why I'm suspicious that something is amiss. If the air gap wasn't there, Vmax would become 8e6 X d and C would be a maximum giving Q=CV the highest possible value. However, once the air gap is introduced, Vmax drops substantially. As the air gap is increased, Vmax increases, but C falls accordingly, and Q seems to be constant as a result. In my attempts at working on this, I recognized that the voltage gradient (delta V / delta d) is 300 times lower in the StontiumTi than in the air, but that the voltage is the gradient X thickness of the layer. So I think I did all that part correctly.
-gt-