Class 6. Capacitance and Capacitors. Physics 106. Winter Press CTRL-L to view as a slide show. Class 6. Physics 106.

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1 and in and Energy Winter 2018 Press CTRL-L to view as a slide show.

2 From last time: The field lines are related to the field as follows: What is the electric potential? How are the electric field and the electric potential related? How can we find the electric field and the electric potential? How are electric fields and electric potentials used in practical applications? The electric potential is the potential energy divided by the charge The electric potential is also called the voltage Applying fields to a CRT in and Energy

3 Today we will discuss: in series and parallel circuits in and Energy

4 in and Energy

5 A capacitor consists of two conductors, one with a charge +Q and one with a charge Q. Often the conductors are parallel plates. The voltage difference between the conductors is V. Out of tradition and laziness, we usually write the voltage difference as just V. in and Energy

6 C Q V Units: Farad (F) 1 F = 1 C / 1 V A farad is very large Often will see µf or pf in and Energy

7 Q = CV A big capacitor holds a large charge at a small voltage. in and Energy

8 in and Energy

9 First developed by Pieter van Musschenbroek in Leyden in 1746 in and Energy

10 Parallel-Plate Capacitor The capacitance of a device depends on the geometric arrangement of the conductors For a parallel-plate capacitor whose plates are separated by air: in and Energy

11 Parallel-Plate Capacitor The capacitance of a device depends on the geometric arrangement of the conductors For a parallel-plate capacitor whose plates are separated by air: C = ɛ 0 A d in and Energy

12 Parallel-Plate Capacitor The capacitor consists of two parallel plates Each have area A They are separated by a distance d The + charge on one plate holds the charge on the other plate in place. in and Energy

13 Parallel-Plate Capacitor If the plates are large, the capacitor can hold more charge. If the plates are closer together, the capacitor can hold more charge, because the + charge attracts the charge more strongly. in and Energy

14 Parallel Plate Capacitor Consists of two conducting plates, one positive and one negative Charge is pulled to the inside surface of either plate The field outside either plate is zero in and Energy

15 Parallel Plate Capacitor Consists of two conducting plates, one positive and one negative Charge is pulled to the inside surface of either plate The field outside either plate is zero in and Energy

16 Electric Field in a Parallel-Plate Capacitor in and Energy The electric field between the plates is quite uniform

17 Example 1: Derive the Parallel-Plate Capacitor Equations Consider a parallel-plate capacitor with plates of area A separated by distance d. The electric field in a capacitor is in and Energy E = σ ɛ 0

18 Example 1: Derive the Parallel-Plate Capacitor Equations The surface charge density is the total charge of a plate divided by its total area: σ = Q A in and Energy This gives: E = σ ɛ 0 = Q ɛ 0 A

19 Example 1: Derive the Parallel-Plate Capacitor Equations The electric field can be given in terms of the voltage: E = V x The sign gives the field direction, but we re only interested in magnitude, so we ignore it. We also write V as V, giving us: in and Energy E = V d So the voltage is V = Ed = Qd ɛ 0 A

20 Example 1: Derive the Parallel-Plate Capacitor Equations The capacitance can then be found: C = Q V = Qɛ 0A Qd in and Energy

21 Example 1: Derive the Parallel-Plate Capacitor Equations The capacitance can then be found: C = Q V = ɛ 0A d in and Energy C = ɛ 0A d

22 A Short Problem: You have two square plates 1.00 m on each side and you wish to make a 1.00 F capacitor. (That s a huge capacitace!) If there is air between the plates, what is the separation distance? in and Energy

23 A Short Problem: You have two square plates 1.00 m on each side and you wish to make a 1.00 F capacitor. (That s a huge capacitace!) If there is air between the plates, what is the separation distance? C = ɛ 0A d d = d ɛ 0A C = m in and Energy

24 A Short Problem: You have two square plates 1.00 m on each side and you wish to make a 1.00 F capacitor. (That s a huge capacitace!) If there is air between the plates, what is the separation distance? in and Energy C = ɛ 0A d d = d ɛ 0A C That s much smaller than one atom! = m

25 in in and Energy

26 in The simplest capacitor circuit is a capacitor connected to a battery with a switch to allow current to flow. in and Energy This is schematically represented as:

27 in When we close the switch, charge flows from the battery into capacitor. in and Energy As the capacitor charges, it pushes charges in the wire in opposition to the battery.

28 in When the voltage on the capacitor matches the voltage of the battery, current ceases to flow. in and Energy The charge on the capacitor is Q = CV where V is the voltage of the battery.

29 Series and Parallel The most common ways of connecting multiple circuit elements are in "series" and "parallel" in and Energy Two capacitors joined in series. Two capacitors joined in parallel.

30 in Parallel The total charge is equal to the sum of the charges on the capacitors Q total = Q 1 + Q 2 The voltage across each capacitor is the same and the same as the voltage of the battery in and Energy

31 Combining in Parallel The capacitors are equivalent to a single capacitor with a capacitance of C eq Q eq = Q 1 + Q 2 in and Energy C eq V = C 1 V + C 2 V C eq = C 1 + C 2

32 Combining in Parallel Adding capacitance in parallel is analogous to increasing the area of a capacitor. in and Energy

33 Combining in Parallel C eq = C 1 + C 2 + The equivalent capacitance of a parallel combination of capacitors is greater than any of the individual capacitors in and Energy

34 in Series Consider the two capacitors shown below. in and Energy

35 in Series Consider the two capacitors shown below. in and Energy As + charge enters on the left, it drives charge from the right plate of the left capacitor to the left plate of the right capacitor.

36 in Series Consider the two capacitors shown below. in and Energy The charge on each capacitor is the same.

37 in Series V eq = V 1 + V 2 Q C eq = Q C 1 + Q C 2 1 C eq = 1 C C 2 in and Energy An equivalent capacitor can be found that performs the same function as the series combination The potential differences add up to the battery voltage

38 in Series V eq = V 1 + V 2 Q C eq = Q C 1 + Q C 2 1 C eq = 1 C C 2 in and Energy An equivalent capacitor can be found that performs the same function as the series combination The potential differences add up to the battery voltage

39 in Series V eq = V 1 + V 2 Q C eq = Q C 1 + Q C 2 1 C eq = 1 C C 2 in and Energy An equivalent capacitor can be found that performs the same function as the series combination The potential differences add up to the battery voltage

40 in Series V eq = V 1 + V 2 Q C eq = Q C 1 + Q C 2 1 C eq = 1 C C 2 in and Energy An equivalent capacitor can be found that performs the same function as the series combination The potential differences add up to the battery voltage

41 in Series Adding capacitors in series is analogous to increasing the distance between capacitor plates. The equivalent capacitance of a series combination is always less than any individual capacitor in the combination in and Energy

42 Energy Stored in a Capacitor Energy stored is U = 1 2 CV 2 From the definition of capacitance, this can be rewritten in different forms: U = 1 2 CV 2 = Q2 2C = 1 2 QV in and Energy

43 in and Energy

44 with A dielectric is an insulating material placed between the plates of a capacitor increase capacitance C = κc 0 = κɛ 0 (A/d) where κ is the dielectric constant include rubber, plastic, or waxed paper in and Energy

45 Dielectric Strength For any given plate separation, there is a maximum electric field that can be produced in the dielectric before it breaks down and begins to conduct This maximum electric field is called the dielectric strength in and Energy

46 with in and Energy Adding a dielectric between charged capacitor plates reduces the voltage. Why?

47 An Atomic Description of Polarization occurs when there is a separation between the negative charge and the positive charge of the dielectric The dielectric becomes polarized because it is in an electric field of the plates in and Energy

48 Adding a Dielectric to a Capacitor with Fixed Charge The charge on the dielectric creates a field that opposes the field of the plates in and Energy

49 Adding a Dielectric to a Capacitor with Fixed Charge The charge on the dielectric creates a field that opposes the field of the plates This reduces the total electric field and the voltage The capacitance therefore increases in and Energy

50 and Current in and Energy

51 We are going to make a human model of circuits Traditionally, we think of positive charge as moving in a circuit. You will be the positive charge. A few of you will be a neutral wire. Hold your hands up and repel each other. Now be a positively-charged wire. Now be a negatively-charged wire. If you were charges on a real wire, where would you go? in and Energy

52 A Battery A batery pushes charges onto one end of a wire and pulls charges off the other end. A few of you will be a battery behind the stand. Now make a current flow around the stand. Where is the wire positive, neutral, negative? Where is the energy of positive charges highest along the wire? What kind of energy is it? Where is the voltage highest in the wire? How do you think electrons actually move in a wire? in and Energy

53 A Resistor A few of you will be a resistor. Charges collide with atoms in the resistor and change their direction of flow. Be a large resistor. Be a small resistor. How does resistance affect current.? What could we do to get more current? in and Energy

54 A Resistor, Part B A few more students will make the resistor longer. What happens to the current? Now make the resitor shorter and wider. What happens to the current? What can you say about the voltage on the wire? in and Energy

55 A Capacitor The battery is now disconnected. But don t go away. A few of you will make a capacitor in front of the stand. Remember that you are replled by positive charges on the other plate, but attracted to the "negative protons" in your own metal plate. Now we ll hook up the battery. What happens to the total charge? What happens to the charge on each plate? What happens to the current? What happens to the voltage? in and Energy

56 Parallel Disconnect the battery again. Now split the capacitor into two separate capacitors, side by side. You will also need some extra wire. These capacitors are in parallel. Reconnect the battery. What can you say about the voltages across the two capacitors? What can you say about the charges on the capacitors? What happens if one capacitor is bigger than the other? in and Energy

57 Series Disconnect the battery Now connect the capacitors in series, one after the other. Reconnect the battery. What happens to the charges between the two capacitors? What can you say about the charge on each capacitor? What can you say about the voltages on the two capacitors? in and Energy