Conservation Laws
Charge and Energy must be conserved! Since changing the value of a capacitor results in a change of voltage and energy, it is difficult to understand what happens to these conserved quantities. Imagine a variable capacitor in which you can alter the distance between the plates? Pulling the plates apart requires work, and this work is responsible for the increase of energy. When you move the plates closer together, you are also doing work. However, pushing the plates closer together, increases the capacitance and the energy decreases. Therefore this argument seems to disagree with the result. It seems to me, that any work done on the capacitor results in the electrons moving to higher or lower energy levels.Balancing the Energy
When an electron accelerates it radiates energy. So any work done on the variable capacitor, will cause an acceleration. The final energy of the capacitor will depend on the amount of work done on it and the amount of energy that is radiated. Click Here For More About How Electrons Lose EnergyInfinite Voltage
Pulling the plates apart increases the voltage. So the question is, if the plates were moved to a separation of infinity, would you would get an infinite voltage? I think that there must be a limit to the amount of voltage that you would gain this way! I believe that the positive plate would attract electrons from somewhere else and break the connection with the negative plate. I personally believe that a separation of charges in a "Thunder Cloud" is responsible for the high voltages of a lightning strike.Some Examples
I will now show some examples of how a fixed charge results in a change of voltage and energy. Since each capacitor is different, there can be no work done on them.Example 1
A charge of 1 coulomb is applied to a 4 farad capacitor.V = q/C = 1/4 volts
E = q2/2C = 1/(2 x 4) = 1/8 joules
Example 2
A charge of 1 coulomb is applied to a 1 farad capacitor.V = q/C = 1/1 = 1 volt
E = q2/2C = 1/(2 x 1) = 1/2 joules
Example 3
A charge of 1 coulomb is applied to a 2 farad capacitor.V = q/C = 1/2 = 1/2 volt
E = q2/2C = 1/(2 x 2) = 1/4 joules
The Energy and Voltage
It is obvious from these examples that for any given charge, the energy needed to hold that charge is different for each capacitor. Also, the value of the capacitor affects the voltage.When Energy is Constant
In the next section, I want to show what happens when the same energy is applied to each capacitor.Example 4
An energy of 1 joule is applied to a 4 farad capacitor.V2 = 2E/ C = 2 x 1 / 4 = 1/2
V = root of 1/2 = 0.7071 volts
q2 = 2 E C = 2 x 1 x 4 = 8
q = root of 8 = 2.8284 coulombs
Example 5
An energy of 1 joule is applied to a 1 farad capacitor.V2 = 2E/ C = 2 x 1 / 1 = 2
V = root of 2 = 1.4142 volts
q2 = 2 E C = 2 x 1 x 1 = 2
q = root of 2 = 1.4142 coulombs
Example 6
An energy of 1 joule is applied to a 2 farad capacitor.V2 = 2E/ C = 2 x 1 / 2 = 1
V = root of 1 = 1 volt
q2 = 2 E C = 2 x 1 x 2 = 4
q = root of 4 = 2 coulombs
Conclusion
When dealing with a fixed value of capacitance, it is clear that for each capacitor there is a different relationship between voltage, charge and energy. However, when dealing with a variable capacitor, this relationship is not so obvious.See also Capacitor Paradox
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