Definition of a Capacitor Special Case: Parallel Plate Capacitor Capacitors in Series or Parallel Capacitor Network Definition of a Capacitor Webassign Chapter 0: 8, 9, 3, 4, 5 A capacitor is a device that is used to store charge. Physically it is two thin plates of metal that are separated by a non-conducting material. Charge is forced onto the plates when they are connected to a battery (or power supply). The amount of charge that is stored on a capacitor is given by the expression: q CV where C is the capacitance of the capacitor, V is the potential of the capacitor, and q is the charge stored on the capacitor. The capacitance (i.e. C ) is a measure of the size of the capacitor. The SI unit of capacitance is the Farad, which is equivalent to a Coulomb / Volt One Farad is a very large capacitance; typical capacitors are measured in µf or nf. If a capacitor is connected directly to a battery, the potential of the capacitor will equal the potential of the batter when the capacitor is fully charged. This expression applies to all capacitors; it is essentially the definition of a capacitor. When a battery pushes charge onto a capacitor, it does work on the charge. The work is done against the repulsive force of the charge that is already stored on the capacitor. Since this force is conservative, potential energy is stored on the capacitor. The amount of potential energy stored is given by: U q C Special Case: Parallel Plate Capacitor Webassign Chapter 0: 9 CV A special case of a capacitor is one in which two metal plates, each of area A are separated by a distance d. The non-conducting material between the plates has a dielectric constant κ. For this special case, the capacitance can be calculated by: qv κεo C A d Page of 5
Capacitors in Series or Parallel Webassign Chapter 0: 0, A circuit diagram of two capacitors connected in series to a battery is: The two capacitors in series have the same charge as the total charge that the battery supplies. And the sum of the potentials of the capacitors must equal the potential of the battery. We can express these ideas algebraically as: q TOT q q V TOT V + V The direct result of these two ideas is that the two capacitors in series behave as if they are one capacitor with a total capacitance given by: C C TOT C + C C Capacitors in Series A circuit diagram of two capacitors connected in parallel to a battery is: The two capacitors in parallel have the same potential as the total potential (i.e. that of the battery.) And the sum of the charge on the capacitors must equal the total charge supplied by the battery. We can express these ideas algebraically as: q TOT q + q V TOT V V The direct result of these two ideas is that the two capacitors in series behave as if they are one capacitor with a total capacitance given by: C TOT + Capacitors in Parallel C C Page of 5
Capacitor Network Webassign Chapter 0: A capacitor network is a circuit that includes three or more capacitors. The capacitors are typically connected, through a combination of series and parallel combinations, to one battery. A capacitor network problem will usually ask you to find the total capacitance of the network, and to find the charge and/or potential for some or all of the individual capacitors. Regardless of what you are asked to find, the procedure for handling a capacitor network problem is always the same. Consider the capacitor network from Problem 5 of Homework : The first step is to disconnect the battery from the network. We do this by removing the battery from the diagram and introducing points a and b : We then redraw the diagram so that point a is at the top, point be is at the bottom, and all capacitors are on vertical lines. (Note that C and C are on horizontal lines in the original diagram.) To redraw, we start at point a and recognize that the first thing we encounter is C. After this, we encounter a branch point. We will draw the branch point so that it branches left and right, with each branch a vertical line. We can now combine the capacitors, two at a time, until only one total capacitor remains. To do this:. Branches are in parallel with each other; if each branch has one capacitor, combine them as capacitors in parallel. Page 3 of 5
. If a branch has more than one capacitor, they are in series on that branch. Combine capacitors on one branch as capacitors in series. 3. Redraw the network at each step, i.e. after each time you combine two capacitors. Following these steps, we recognize that C and C 4 are in series on the right branch. We must first combine these using Rule before we can use Rule to combine the right and left branch. (Note: calculate the value of C 4 using the expression for capacitors in series.) Now that each branch has only one capacitor, the next step is to combine C 3 and C 4 in parallel. (Note: calculate the value of C 34 using the expression for capacitors in parallel.) Finally, the remaining two capacitors are in series between points a and b. Combine them and calculate C TOT using the expression for capacitors in series. The final diagram is: Page 4 of 5
At this point you have found the total capacitance or equivalent capacitance of the network. We can now find the charge and potential of each individual capacitor in the network. This requires that we reconnect the battery to points a and b, although we do not have to show this in the diagram. It is easier to just acknowledge that there is a battery now connected to these points. If we imagine connecting the battery to the final diagram, then then potential of the battery can be labeled V TOT ; that is, it is the total potential of the network. Using C TOT and V TOT, calculate q TOT Note that q TOT is used just as a means to figure out the q and V for the individual capacitors. It is a crucial step in this process. Now work your way backward through the diagrams. The procedure is:. Ask where did C TOT come from? The answer will always be two capacitors from the previous diagram that were either in series or parallel.. If C TOT came from two capacitors in series, then the two individual capacitors have the same charge as C TOT. Translate the value of q TOT to the individual capacitors, then calculate V for each of these individual capacitors. 3. If C TOT came from two capacitors in parallel, then the two individual capacitors have the same potential as C TOT. Translate the value of V TOT to the individual capacitors, then calculate q for each of these individual capacitors. 4. Once you have q and V for the two capacitors that you combined to create C TOT, repeat this process by applying Step to the capacitor that was created in the second to last diagram (e.g. C 34 in our Problem 5.) Using this process, you should end up at the initial diagram with q and V for each of the individual capacitors. Page 5 of 5