PHY 184

1. PHY 184 Spring 2007 Lecture 11 Title: Capacitors 184 Lecture 11

2. Announcements • Homework Set 3 is due Tuesday, Jan. 30 at 8:00 am. • Homework Set 4 opened this morning. • Today we will finish up the electric potential and start with capacitors. 184 Lecture 11

3. Review - Electric Potential, V(x) ► If a charge q moves in an electric field… ► For reference V=0 at infinity… ► E is the gradient of V… 184 Lecture 11

4. Review - Electric Potential (2) ► For a point source Q … ► For many sources … (superposition) 184 Lecture 11

5. Problem-Solving Strategies Note: • Given a charge distribution, calculate the electric field and the electric potential: (i) Use Gauss Law to derive the electric field E: (ii) For the potential (V=0 at infinity) use • For an arrangement of charges, remember the superposition principle! This holds for the electric force, the electric field, the electric potential and the electric potential energy. 184 Lecture 11

6. Capacitors 184 Lecture 11

7. Capacitors • Capacitors are devices that store energy in an electric field. • Capacitors are used in many every-day applications • Heart defibrillators • Camera flash units • 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. 184 Lecture 11

8. Capacitance • Capacitors come in a variety of sizes and shapes. • Concept: A capacitor consists of two separated conductors, usually called plates, even if these conductors are not simple planes. • We will define a simple geometry and generalize from there. • We will start with a capacitor consisting of two parallel conducting plates, each with area A separated by a distance d . • We assume that these plates are in vacuum (air is very close to a vacuum). 184 Lecture 11

9. Parallel Plate Capacitor • We charge the capacitor by placing • a charge +q on the top plate • a charge -q on the bottom plate • Because the plates are conductors, the charge will distribute itself evenly over the surface of the conducting plates. • The electric potential, V, is proportional to the amount of charge on the plates. e.g., using a battery More precisely, potential difference V(+)-V(-) = V 184 Lecture 11

10. Parallel Plate Capacitor (2) • The proportionality constant between the charge q and the electric potential difference V is the capacitance C. • We will call the electric potential difference V the “potential” or the “voltage” across the plates. • The capacitance of a device depends on the area of the plates and the distance between the plates, but does not depend on the voltage across the plates or the charge on the plates. • The capacitance of a device tells us how much charge is required to produce a given voltage across the plates. 184 Lecture 11

11. Clicker Question • What is the NET CHARGE on the charged capacitor? A: +q+(-q)=0 B: |+q|+|-q|=2q C: q D: none of the above 184 Lecture 11

12. Clicker Question What is the NET CHARGE on the charged capacitor? A: +q+(-q)=0 Charges are added with their signs. However, we refer to the charge of a capacitor as being q (the charge of a capacitor is not the net charge!) 184 Lecture 11

13. Definition of Capacitance • The definition of capacitance is • The units of capacitance are coulombs per volt. • The unit of capacitance has been given the name farad (abbreviated F) named after British physicist Michael Faraday (1791 - 1867) • A farad is a very large capacitance • Typically we deal with F (10-6 F), nF (10-9 F),or pF (10-12 F) 184 Lecture 11

14. Charging/Discharging a Capacitor • We can charge a capacitor by connecting the capacitor to a battery or to a DC power supply. • A battery or DC power supply is designed to supply charge at a given voltage. • When we connect a capacitor to a battery, charge flows from the battery until the capacitor is fully charged. • If we then disconnect the battery or power supply, the capacitor will retain its charge and voltage. • A real-life capacitor will leak charge slowly, but here we will assume ideal capacitors that hold their charge and voltage indefinitely. 184 Lecture 11

15. Charging/Discharging a Capacitor (2) • Illustrate the charging processing using a circuit diagram. • Lines represent conductors • The battery or power supply is represented by • The capacitor is represented by the symbol • This circuit has a switch (ab) • (open) When the switch is between positions a and b, the circuit is open (not connected). • (pos a) When the switch is in position a, the battery is connected across the capacitor. Fully charged, q = CV. • (pos b) When the switch is in position b, the two plates of the capacitor are connected. Electrons will move around the circuit – a current will flow -- and the capacitor will discharge. 184 Lecture 11

16. Demo: Big Spark • 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! 184 Lecture 11

17. Parallel plate capacitor – -- a simple ideal model 184 Lecture 11

18. Parallel Plate Capacitor • Consider two parallel conducting plates separated by a distance d • This arrangement is called a parallel plate capacitor. • The upper plate has +q and the lower plate has –q. • The electric field between the plates points from the positively charged plate to the negatively charged plate. • We will assume ideal parallel plate capacitors in which the electric field is constant between the plates and zero elsewhere. • Real-life capacitors have fringe field near the edges. 184 Lecture 11

19. Parallel Plate Capacitor (2) We can calculate the electric field between the plates using Gauss’ Law We take a Gaussian surface shown by the red dashed line Flux (through bottom surface of the top plate only!) = EA Enclosed charge = q 184 Lecture 11

20. Parallel Plate Capacitor (3) • Now we calculate the electric potential across the plates of the capacitor in terms of the electric field. • We define the electric potential across the capacitor to be V and we carry out the integral in the direction of the blue arrow. Note: V = V(+) - V(-) V = Ed 184 Lecture 11

21. Parallel Plate Capacitor (4) Remember the definition of capacitance… … so the capacitance of a parallel plate capacitor is Variables: A is the area of each plate d is the distance between the plates Note that the capacitance depends only on the geometrical factors and not on the amount of charge or the voltage across the capacitor. 184 Lecture 11

22. Demo: Parallel Plate Capacitor • The voltage of a charged capacitor as function of the distance between the plates • For constant charge, the voltage is proportional to the distance between the plates 184 Lecture 11

23. Example - Capacitance of a Parallel Plate Capacitor • We have a parallel plate capacitor constructed of two parallel plates, each with area 625 cm2 separated by a distance of 1.00 mm. • What is the capacitance of this parallel plate capacitor? C = 0.553 nF Result: A parallel plate capacitor constructed out of square conducting plates 25 cm x 25 cm separated by 1 mm produces a capacitor with a capacitance of about 0.5 nF. 184 Lecture 11

24. 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 1.00 F? Result: A parallel plate capacitor constructed out of square conducting plates 10.6 km x 10.6 km (6 miles x 6 miles) separated by 1 mm produces a capacitor with a capacitance of 1 F. 184 Lecture 11

25. Example - Capacitance, Charge and Electrons … • A storage capacitor on a random access memory (RAM) chip has a capacitance of 55 nF. If the capacitor is charged to 5.3 V, how many excess electrons are on the negative plate? Idea: We can find the number of excess electrons on the negative plate if we know the total charge q on the plate. Then, the number of electrons n=q/e, where e is the electron charge in coulomb. Second idea: The charge q of the plate is related to the voltage V to which the capacitor is charged: q=CV. 184 Lecture 11