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Capacitors (Energy (The energy stored in a capacitor is equal to the work…
Capacitors
Energy
The energy stored in a capacitor is equal to the work done in moving the stored charge between the capacitor plates, against the potential difference.
This equals the magnitude of the stored charge, multiplied by the average potential difference between the plates, as the capacitor plates charge.
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Capacitance \(C\)
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\[C=\frac{Q}{V}\]
Where \(V\) is the potential difference between the plates and \(Q\) is the magnitude of the charge stored on each plate
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Factors
Separation of Plates
As the plates are brought closer together the electric field strength between the plates will increase, as for a uniform electric field \(E=\frac{V}{d}\). This means that more charge can be repelled from the positive plate by the negative plate, for a given potential difference. More charge can accumulate on the plates and the capacitance increases.
Surface Area of Plates
The greater the volume of the plates the more charge can fit onto the plates and so \(Q\propto Volume\). If the depth of the plates is small compared to the area, then it is negligible and the charge stored is proportional to the area of the plates. This means that the capacitance increases with plate surface area.
Permittivity of Dielectric Medium
The greater the permittivity of the medium between the plates the lower the electric field strength between the plates for a given amount of accumulated charge. This lowers the potential difference between the plates as for a uniform electric field \(E\propto V\). This means that there will be a lower potential difference for a given amount of charge, and so the capacitance will have increased.
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Dielectrics
When an electric field is set up in a polar medium, the molecules that make up the medium will line up with the electric field.
The molecules align in anti parallel, due to the attraction of opposite charges. The polar molecules then create a second electric field (due to their common alignment) which counters the first, and so reduces the overall electric field strength.
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Absolute Permittivity
A measure of the amount of charge needed to generate an electric field of a given size in a material
Relative Permittivity
The ration of the permittivity of a medium to the permittivity of free space
\[\epsilon_{r}=\frac{\epsilon_{1}}{\epsilon_{0}}\]
Charging and Discharging
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Time Constant \(RC\) or \(\tau\)
Of a discharge circuit is equal to the time is would take the capacitor in the circuit to discharge to 37% of it's initial charge, or charge to 63% of it's total charge capacity
Time to Half \(T_{\frac{1}{2}}\)
The time it takes the charge on a discharging capacitor to reach half it's initial value
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It takes roughly 5 time constants for a capacitor circuit to fully charge or fully discharge from full
Uses
Capacitors can be used in smoothing DC supplies, by charging when the potential difference becomes high, and discharging when the potential difference becomes smaller.
Ultra capacitors can store more charge than regular capacitors, and so can be used to back up large power supplies for a short period of time
Capacitors have a high power density, meaning that they can supply a large amount of energy in a short space of time. This makes then useful in camera flashes.