PES 1120 Spring 2014, Spendier

Lecture 16/Page 1

Today: - Review last time - Capacitors in series and parallel - Capacitors in Circuits Capacitors in Circuits Last lecture comment: Shouldn’t we connect the parallel-plate capacitor in a loop? Yes! We need to start to talk about circuits. Capacitors are used in circuits extensively. Charging a capacitor:

The positive terminal of the battery has a higher potential than the negative terminal. In this respect the battery stores electrical energy by maintaining a potential difference via electrochemical reactions. The battery does work in order to put charge on the plates on the capacitor. The capacitor stores energy through the potential difference due to the stored charge on thee plates. The energy in the capacitor can be released very quickly by “discharging” it, whereas the battery releases energy slowly. What happens when the switch in the diagram is closed? When the switch is closed, the capacitor begins to accumulate charge. Before this the plates have no net charge. Electrons (the only charges that can move through the wires) are attracted from the left plate l to the positive terminal of the battery  l builds up a positive charge Electrons are also repelled from the negative terminal of the battery and accumulate on the right plate r  plate r builds up a negative charge. When V(- terminal) = V(r) And (V+ terminal) = V(l) Then there is no more force causing the electrons to move. (Electrons only move to decrease their potential energy.) Last lecture we learned that the amount of charge that can accumulate on the plate is given by Q = C (Vl-Vr)

PES 1120 Spring 2014, Spendier

Lecture 16/Page 2

Obviously a larger battery can supply more charge.

C….capacitance units [C/V] = [F] Farad C

How much ch arg e stored Q  Re lated to work done to chrage plates Vab

C  Unit     F ....( Farad )  V 

Last lecture we calculated capacitance for different geometries:

eA Q  0 Vab d rr Spherical capacitor: C  4pe0 a b (made mistake last lecture, please check!!!!) rb  ra Parallel-plate capacitor: C 

Vab = Va - Vb = potential of the positively charged conductor a with respect to the negatively charged conductor b Question last class – which geometry is more efficient in storing energy? Let’s do an example:

PES 1120 Spring 2014, Spendier

Lecture 16/Page 3

Example 1: The plates of a spherical capacitor have radii 38.0 mm and 40.0 mm a) Calculate the capacitance b) What must be the plate area of a parallel-plate capacitor with the same plate separation and capacitance? c) Now compare area found in b) to surface area for sphere...

c)

ra2

Changing Capacitance Since a single capacitor has such a small capacitance, we are often interested in ways of changing its value. To change the capacitance value: (1) Different shape plates (spherical versus parallel-plate capacitor see above). (2) Connecting individual capacitors together. (3) Inserting an insulator between the plates. The remainder of this lecture we will focus on (2) Capacitor Combinations Combinations of capacitors behave like a single equivalent capacitor, Ceq. Why do we need to understand this? Capacitors are manufactured with certain standard capacitances and working voltages. However, these standard values may not be the ones you actually need in a particular application. You can obtain the values you need by combining capacitors; many combinations are possible, but the simplest combinations are a series connection and parallel connection.

PES 1120 Spring 2014, Spendier

Lecture 16/Page 4

Connect in Parallel:

–Have the same potential V1 = V2 = V –Charge depends on capacitance (Charge redistributes between capacitors - think about water in a pipe - when it comes to an intersection of two pipes the water has to divide between both) Q = Q1 +Q2 or Q1 = C1V Q2 = C2V Q Q1  Q2 Q1 Q2     C1  C2 V V V V This works no matter how many capacitors we add in series. For n capacitors connected in series: Ceq 

n

Ceq   Ci i 1

The equivalent capacitance of a parallel combination equals the sum of the individual capacitances. In parallel connection the equivalent capacitance is always greater than any individual capacitance.

PES 1120 Spring 2014, Spendier

Lecture 16/Page 5

Capacitors in Series

Connect in Series: – Have the same charge Q1 = Q2 = Q – Potential difference across them add V1 +V2 = V Why? V  Va  Vb  Va  Vc   Vc  Vb   V1  V2 If you wanted to replace these capacitors with just one equivalent capacitor:

Q Q  V V1  V2 1 V  V2 V1 V2  1   Ceq Q Q Q

Ceq 

1 1 1   Ceq C1 C2 This works no matter how many capacitors we add in parallel. For n capacitors connected in parallel: n 1 1  Ceq i 1 Ci

The reciprocal of the equivalent capacitance of a series combination equals the sum of the reciprocals of the individual capacitances. In a series connection the equivalent capacitance is always less than any individual capacitance.

PES 1120 Spring 2014, Spendier

Lecture 16/Page 6

Example 2: Let C1 = 6.0 μF, C2 = 3.0 μF, and Vab = 18 V. Find the equivalent i) capacitance, and find ii) the charge and iii) potential difference for each capacitor when the two capacitors are connected (a) in series and (b) in parallel.

PES 1120 Spring 2014, Spendier

Lecture 16/Page 7

Capacitor Circuits When there are combinations of series and parallel capacitors, try to reduce them to more simple forms. Example 3: Find the equivalent capacitor of a)