CH 24: capacitors

1. CH 24: capacitors

2. What happens when we bring two oppositely charged conductors close to each other? • An electric field is directed between the two conductors from positive to negative. • The presence of the electric field requires that there be an electric potential difference between the two conductors. • Therefore the amount of excess charge on each of the conductors is directly related to the potential difference between the two conductors. The more charge the greater the electric field strength and hence the greater the electric potential. It could be surmised that if we had an external source of charge and we increased the electric potential difference between the two conductors we should be able to increase the amount of charge on each of the conductors. This suggests that by applying a potential difference between two conductors we can store charge on each of the conductors. This means that conductors will have a certain amount of charge stored on them for a specified electric potential difference. This means that conductors have a certain capacity for storing charge. We call this Capacitance. Capacitance – The amount of charge that can be stored on conductor at a particular potential difference.

3. We can therefore mathematically define the capacitance as: C – Capacitance [F] Q – Total charge stored on each capacitor [C] DV – Electric potential difference between the capacitors [V] F = Farad – This is the amount of charge stored per volt of potential difference. A single conductor could also have capacitance. The potential of a single conductor is usually referenced to a point that has zero potential, which is located an infinite distance away. The capacitance of a spherical conductor can be determined in the following way. 0 Radius of the charged conductor Electric potential at surface of a point charge with radius R (acceptable when looking outside of sphere)

4. The capacitance shows no dependence on the charge and the electric potential. The electric potential can determine the amount of charge on a conductor (or vise versa), but is restricted by the capacitance of the conductor. What then does the capacitance depend on? To answer this let us look at the simple case of a pair of parallel conducting sheets of a specified area A and specified separation distance d. The electric field from a charged conductor is: This is the electric field between the two parallel plates The electric potential is related to the electric field and the separation distance This is the magnitude of the potential difference between the parallel plates when E is uniform and parallel to d. This is the electric potential difference for this pair of parallel conducting plates. For a conductor

5. Let us now determine the capacitance for this set of parallel conducting plates. Notice that the capacitance only depends on the physical geometry of the pair of conductors. In order to change the capacitance you must alter the area or separation distance. Yes, this does make sense for our definition of capacitance. Does this make sense? How does the area of the plates affect the amount of charge that can be stored? The greater the area the more space there is to pack in electrons, and therefore the greater total amount of charge that can be contained on the surface of the conductor. How does the separation distance affect the amount of charge that can be stored? If the separation distance is decreased, the electric potential between the two plates would decrease. The external voltage source sets the electric potential to a constant value, and therefore more charge would need to be added to the plates to compensate for the decreased separation distance. The additional charge increases the electric field strength and hence the electric potential between the plates returning it to the value it had prior to the decrease in the separation distance.

6. What could our source of charge be? A typical source of charge would be a battery. A simple battery is an electrochemical cell where two metals placed in an electrically conductive fluid generate excess electrons (or an absence of electrons) due to surface chemical reactions. An electric potential difference exists between the two metals in the electrolyte, which corresponds to the voltage measured across the terminals of the battery A chemical reaction will generate a constant electric potential difference until the reactants are “used up”. This will be observed as a decrease in the electric potential difference.

7. Consider a capacitor made of two parallel metallic plates separated by a distance d. The top plate has a surface charge density +s, the bottom plate –s. A slab of metal of thickness l< dis inserted between the plates, not connected to either one. Upon insertion of the metal slab, the potential difference between the plates 1. increases. 2. decreases. 3. remains the same. C increases because d decreases DV decreases because C increases