Capacitor Paradox

The Capacitor Paradox is about charging one capacitor from another and observing how charge and energy behave.

C1 and C2 both have a value of 1 farad.  C1 is charged to a voltage of 2 volts, and C2 is completely discharged.

Charge q = cv

So the charge on C1 = 1 x 2 = 2 coulombs

Energy = 1/2 c v ²

So the energy of C1 = 1/2 x 1 x 2 ² = 2 joules

Closing the Switch

When the switch is closed, the charge is shared between both capacitors.  Now the circuit has a capacitance of 2 farads, and a voltage of 1 volt.

q = cv = 2 x 1 = 2 coulombs (as before)

E = 1/2 c v ² = 1/2 x 2 x 1 ² = 1 joule

Energy is Missing

One joule of energy is missing.  Since energy is a conserved quantity it needs to be accounted for.  The reason for this discrepancy is that when electrons accelerate, they radiate energy. Click Here For More About Electrons Losing Energy

Charge

Charge is related to the number of electrons either flowing or stored.  Charge in this case refers to the number of electrons stored.  In a capacitor, excess electrons are held on the negative plate and the positive plate contains atoms with the same number of missing electrons.  Until those electrons get restored to there atoms, a voltage will exist between the plates.

Energy and Voltage

If a large voltage is applied to an electron, it will move quickly and therefore have more kinetic energy.  So more volts means more energy.  In the case of the energy stored by a capacitor, the energy is Potential Energy, which converts to kinetic energy when the capacitor is discharged.

Capacitance and Energy

Consider charging C1 and C2 separately to 1 volt each.

E_c1 = 1/2 x 1 x 1 ² = 1/2 Joules

E_c2 = 1/2 x 1 x 1 ² = 1/2 Joules

Total energy = 1 Joule

The result is the same as when C2 was charged by C1.  Since they were charged separately, there can be no loss of energy.  However, when C2 was charged by C1, electrons had to flow around the circuit.  This movement allowed the electrons to radiate the excess energy.  What this example illustrates is that 1 joule is the correct value!

Where does the Energy Go?

Energy = 1/2 c v ²

Charge q = cv

Therefore E = q² / 2C

So increasing the capacitance, reduces the amount of energy needed to hold a given charge.  When less energy is needed, the excess energy is radiated by the movement of the electrons.  i.e. Movement causes the electrons to accelerate and when an electron accelerates it radiates energy.

Force and Energy

In a large capacitor, the charges are spread out more than in a small one.  So its a bit like people in a lift (elevator).  If the lift is small, people are close together, so they want to get out quickly.  So the small capacitor will give more energy to the charges, because there is a greater force of repulsion.  In other words, the electrons on the negative plate are closer together and push harder against one another.

Closer Together

Now lets consider the separation of the plates of a capacitor.  If the plates are close together, the electrons are held in place by the attraction of the positive charges.  So they will want to stay in place, whereas when the plates are further apart, they will want to get away.

Summing Up

Changing the capacitor value, changes the forces between the charges.  Since force and energy are related, the energy changes accordingly.  Therefore, a change of capacitance, causes a change of energy.  However, increasing the value of capacitance causes the electrons to move.  This movement causes the electrons to radiate the excess energy.  Decreasing the capacitance requires work to be applied to the capacitor plates, so this work gives energy to the electrons.  In the case of the capacitor paradox, electrons were made to flow and therefore radiate the excess energy.