Capacitors. February 2, 2014 Physics for Scientists & Engineers 2, Chapter 24 1

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1 Capacitors February 2, 2014 Physics for Scientists & Engineers 2, Chapter 24 1

2 Notes! Exam 1 was a success 81% average score including correction set! First score update was sent out over lon-capa HW 1 %, exam 1 % (including correction set), clicker % If your clicker score is zero, I don t have you clicker number said HW1 and HW2, but actual HW score only included HW1 February 2, 2014 Physics for Scientists & Engineers 2, Chapter 24 2

3 Capacitors! Capacitors are devices that store energy in an electric field! Capacitors are used in many every-day applications Heart defibrillators Camera flash units Touch screens! Capacitors are an essential part of electronics! Capacitors can be micro-sized on computer chips or super-sized for high power circuits such as FM radio transmitters and exist in a variety of shapes February 2, 2014 Physics for Scientists & Engineers 2, Chapter 24 3

4 Capacitance! A capacitor consists of two separated conductors, usually called plates, even if these conductors are not simple planes! If we take apart a typical capacitor, we might find two sheets of metal foil separated by an insulating layer of Mylar! The sandwiched layers can be rolled up with another insulating layer into a compact form that does not look like parallel sheets of metal February 2, 2014 Physics for Scientists & Engineers 2, Chapter 24 4

5 Capacitance! Assume a convenient geometry and then generalize the results! The geometry we choose is a parallel plate capacitor, which consists of two parallel conducting plates, each with area A, separated by a distance d, in a vacuum! The capacitor is charged by placing a charge of +q on one plate and q on the other plate! The electric potential, ΔV, between the plates is proportional to the amount of charge on the plates February 2, 2014 Physics for Scientists & Engineers 2, Chapter 24 5

6 Capacitor potential! Because the plates are conductors, the charge will distribute itself uniformly over the plates! We can use the techniques in Chapter 23 to calculate the potential numerically using a computer! The potential has a steep drop between the plates and a gradual drop outside the plates! Thus the electric field will be strong between the plates and weak outside the plates February 2, 2014 Physics for Scientists & Engineers 2, Chapter 24 7

7 Capacitor potential! We can take a slice through the x-y plane! The equipotential lines are close together between the plates and far apart outside the plates February 2, 2014 Physics for Scientists & Engineers 2, Chapter 24 8

8 E r ( ) = V ( r )! Here we show the electric field vectors at regularly spaced grid points in the x-y plane! The field between the plates is perpendicular to the plates and has a much larger magnitude than the field outside the plates! The field outside the plates is the fringe field Capacitor field February 2, 2014 Physics for Scientists & Engineers 2, Chapter 24 9

9 Capacitance! The potential difference between the two plates is proportional to the amount of charge on the plates! The proportionality constant is the the capacitance of the device C = q ΔV! The capacitance of a device depends on the area of the plates and the distance between the plates but not on the charge or potential difference! The capacitance tells how much charge is required to produce a given potential difference between the plates! We can rewrite this equation as q = CΔV C q = V q = VC February 2, 2014 Physics for Scientists & Engineers 2, Chapter 24 10

10 Capacitance! The units of capacitance are coulombs per volt! A new unit was assigned to capacitance, named after British physicist Michael Faraday ( ) called the farad (F) 1 F = 1 C 1 V! One farad represents a very large capacitance! Typical capacitors have capacitances ranging from 1 pf to 1 μf! With this definition, we can write the electric permittivity of free space as ε 0 = F/m February 2, 2014 Physics for Scientists & Engineers 2, Chapter 24 12

11 Circuits! An electric circuit consists of wires that connect circuit elements! These elements can be capacitors! Other important elements include resistors, inductors, ammeters, voltmeters, diodes, and transistors! Circuits usually need a power source! A battery can provide a fixed potential difference commonly called voltage! An AC power source provides a time-varying potential difference February 2, 2014 Physics for Scientists & Engineers 2, Chapter 24 14

12 Circuit Symbols! Circuit elements are represented by commonly used symbols February 2, 2014 Physics for Scientists & Engineers 2, Chapter 24 15

13 Charging and Discharging a Capacitor! A capacitor is charged by connecting it to a battery to create a circuit! Charge flows from the battery to the capacitor until the potential difference across the capacitor is the same as the potential difference across the battery! If the capacitor is then disconnected, it will hold its charge and potential difference! We can use a circuit diagram to illustrate the charging/discharging process Switch position a charges the capacitor Connects the battery across the plates Switch position b discharges the capacitor Shorts the plates together February 2, 2014 Physics for Scientists & Engineers 2, Chapter 24 16

14 Parallel Plate Capacitor! We will consider an ideal parallel plate capacitor! Two parallel conducting plates in a vacuum with charge +q on one plate and q on the other plate! The field is constant between the plates and zero outside February 2, 2014 Physics for Scientists & Engineers 2, Chapter 24 17

15 Parallel Plate Capacitor! We can calculate the field using Gauss s Law E d A = q ε 0! We use the Gaussian surface shown by the dotted red line! We add the contributions to integral from the top, the bottom, and the sides! The sides are outside the capacitor, so the field is zero! The top is inside the conductor, so the field is zero! The bottom part is in the constant field between the plates February 2, 2014 Physics for Scientists & Engineers 2, Chapter 24 18

16 Parallel Plate Capacitor! The electric field is constant and points downward! The vector normal to the surface also points downward! So the integral over the Gaussian surface becomes E da = EdA = E da = EA = q ε 0! The electric potential difference across the two plates is ΔV = i f E d s! The path of integration is chosen to be from the negatively charged plate to the positively charged plate, which gives us ΔV = 0 d ( Eds) = Ed = qd ε 0 A! Combining these equations gives us the capacitance of a parallel plate capacitor C = q ΔV = ε A 0 d February 2, 2014 Physics for Scientists & Engineers 2, Chapter C ε = d 0 A

17 Demo: Large Capacitance! Energy stored in this particular capacitor: 90 J! This is equivalent to the kinetic energy of a mass of 1 kg moving at a velocity of 13.4 m/s! E = 1 2 mv 2 v = 2E m = 2 90 J 1 kg = 13.4 m/s February 2, 2014 Physics for Scientists & Engineers 2, Chapter 24 20

18 Example - Capacitance of a Parallel Plate Capacitor! We have a parallel plate capacitor constructed of two parallel plates, each with area 625 cm 2 separated by a distance of 1.00 mm.! What is the capacitance of this parallel plate capacitor? C = ε 0A d A = 625 cm 2 = m 2 d = 1.00 mm = m ( )( m 2 ) C = F/m m = F C = nf A parallel plate capacitor constructed out of square conducting plates 25 cm x 25 cm (=625 cm 2 ) separated by 1 mm produces a capacitor with a capacitance of about 0.55 nf February 2, 2014 Physics for Scientists&Engineers 2 22

19 Example 2 - Capacitance of a Parallel Plate Capacitor! We have a parallel plate capacitor constructed of two parallel plates separated by a distance of 1.00 mm.! What area is required to produce a capacitance of 0.55 F? C = ε 0 A d d = 1.00 mm = m A = dc ε 0 = ( m) 0.55 F ( ) ( F/m) = m 2 A parallel plate capacitor constructed out of square conducting plates 7.9 km x 7.9 km (4.9 miles x 4.9 miles) separated by 1 mm produces a capacitor with a capacitance of 0.55 F February 2, 2014 Physics for Scientists&Engineers 2 23

Objectives. Capacitors 262 CHAPTER 5 ENERGY

Objectives. Capacitors 262 CHAPTER 5 ENERGY Objectives Describe a capacitor. Explain how a capacitor stores energy. Define capacitance. Calculate the electrical energy stored in a capacitor. Describe an inductor. Explain how an inductor stores energy.