Electric Potential and Capacitance

1. Electric Potential and Capacitance What’s a volt anyway? Presentation 2001 Dr. Phil Dauber as modified by R. McDermott

2. Why Potential? • Electric potential can be visualized as “height”. • This allows us to make comparisons to gravity. • Potential is independent of the object in the electric field.

3. What is Potential? • By agreement from fields, we always define electrical quantities in terms of positive charge. • Positive charges will always move away from other positive charges and toward negative charges, much as a ball always rolls downhill. • By analogy, we consider a region of positive charge to be at a high potential (hill), and a region of negative charge to be at a low potential (valley). • Field lines, then, show the most direct “downhill” path.

4. Equipotential Lines • Maps dealing with hills and valleys are called contour or topographical maps, and consist of closed lines that connect points that have the same height or altitude. • We draw similar “maps” for dealing with charges using Equipotential lines, which connect points that have the same electrical height (potential). • An equipotential line will always be perpendicular to the electric field, since the field always points “downhill”. • Stationary charges will not move on their own along an equipotential line, because that will not lead them “downhill”.

6. Hills and Valleys, Oh My! • As a ball rolls downhill, the gravitational field does work on the ball due to the unbalanced force (weight) acting on the ball, causing it to gain speed (and kinetic energy). We say that potential energy stored in the field between the ball and Earth is converted into kinetic energy. • Similarly, as a positive charge moves from high potential to low the electric field does work on the charge due to the unbalanced force acting on it, causing it to gain speed (and kinetic energy). We say that potential energy stored by the field and charge is converted into kinetic energy.

7. Hills and Valleys, Oh My!

8. Electric and Gravitational Fields Gravity: Electricity: • Property: charge • Property: mass • 1 Sign: positive • 2 Signs: pos, neg • Dependency: 1/r2 • Dependency: 1/r2 • G = 6.67x10-11 n-m2/kg2 • k = 8.99x109 n-m2/C2

9. Electric and Gravitational Fields Gravity: Electricity: • F= (GmEarth/rEarth2)m2 • F = gfieldm2 • F = (kq1/r2)q2 • F = Efieldq2 • PE = mgh • PE = qEx (x = d) • PE is in Joules • PE is in Joules

10. Electric and Gravitational Fields Gravity: Electricity: • We want to treat all objects the same way, so.. • “Potential”, V = Eqd/q = Ed • We want to treat all objects the same way, so.. • “Liftage”, L = mgh/m = gh • L= (PE)/mass • Units = Joules/kg • V = (PE)/charge • Units = Joules/Coul.

11. Electric and Gravitational Fields Gravity: Electricity: • g = F/m • Units = N/kg • g is the grav. field strength • E = F/q • Units = N/c • E is the electric field strength

12. Electric Potential and Electric Field • Can describe charge distribution in terms field or potential. Consider uniform field: • F = Eq • W = qVba • W = Fd =qE d • Thus Vba = Ed or E = Vba/d • Alternate units for E: volts per coulomb

13. Electrical Potential • Potential is potential energy per unit charge • Analogous to field which is force per unit charge • Symbol of potential is V; Va = PEa/q • Only potential differences are measurable; zero point of potential is arbitrary • Vab = Va – Vb = - Wba/q • Wba is work done to move q from b to a

14. Units: • Unit of electric potential is the volt • Abbreviation V • 1 V = 1 Joule/Coulomb = 1 J/C • Thus electrical work = qV If q is in coulombs, the work is in joules; if q is in elementary charges, the work is in electron-volts (eV). • Potential difference is called voltage

15. Electrical Potential Energy • The change in potential energy equals the negative of the work done by the field. • Electrical PE is transferred to the charge as kinetic energy. • A positive charge has its greatest PE near another positive charge or positive plate. • Only differences in potential energy are measurable

16. V = 0 Arbitrary • Usually ground is zero point of potential • Sometimes potential is zero at infinity • + terminal of 12V battery is said to be at 12V higher potential than – terminal

17. Electric Potential and Potential Energy • DPE = PEb – PEa = qVba • If object with charge q moves through a potential difference Vba its potential energy changes by qVba • Example: What is the gain of electrical PE when 1 C of charge moves between the terminals of a 12 volt battery? • Water Analogy: • Voltage is like water pressure (depth or height)

18. Example: Electron in Computer Monitor • An electron is accelerated from rest through a potential difference of 5000 volts • Find its change in potential energy • Find its speed after acceleration Vba = 5000 v = Vb - Va PE = 8x10-16 J V = 4.2x107 m/s a b

19. Questions on Preceding Example • Does the energy depend on the particle’s mass? • Does the final speed depend on the mass?

20. Electric Potential Due to Point Charges • V = k Q/r derived from Calculus • Here V = 0 at r = infinity; V represents potential difference between r and infinity

22. Example: Work to force two point charges together • What work is needed to bring a 2mC charge to a point 10cm from a 5 mc charge? • Work required = change in potential energy • W = qVba = q{kQ/rb – kQ/ra}

23. The Electron Volt • The energy acquired by a particle carrying a charge equal to that of the electron when accelerated through one volt. • W = qV, if q is in Coulombs, then W is in Joules. If q is in elementary charges, W is in electron volts (eV). • 1 eV = 1.6 x 10-19 joules • It’s about ENERGY! • 1 KeV = 1000 eV; 1 MeV = 106 eV • 1 Gev = 109 eV

24. Potential at an Arbitrary Point Near Several Point Charges • Add the potentials due to each point charge • Use the right sign for the charge • Relax; potential isn’t a vector • What is true about the mid-plane between two equal point charges of opposite sign?

25. Capacitors • Store charge • Two conducting plates • NOT touching • May have insulating material between Q = CV

26. Capacitance • Symbol C • Unit: coulombs per volt = farad • 1 pF = 1 picofarad = 10-12 farad • 1nf = 1 nanofarad = 10-9 f • 1mf = 1 microfarad = 1-6 f

27. C is Constant for a Given Capacitor • Does not depend on Q or V • Proportional to area • Inversely proportional to distance between plates • C = e0 A/d • If dielectic like oil or paper between plates use e = Ke0; K is called dielectric constant

29. Find the Capacitance • A capacitor can hold 5 mC of charge at a potential difference of 100 volts. What is its capacitance • C = Q/V = 5 x 10-6 C/ 100V = 5 x 10-8 f = 50000 pf

31. Capacitors Photos courtesy Illinois Capacitor, Inc

32. Applications • In automotive ignitions • In strobe lights • In electronic flash • In power supplies • In nearly all electronics

33. A Capacitor Stores Electric Energy • A battery produces electric energy bit by bit • A capacitor is NOT a type of battery • A battery can be used to charge a capacitor

34. Energy in a Capacitor Holding Charge Q at Voltage V • U = QV/2 = CV2/2 = Q2/2C • Derivation: the work needed to charge a capacitor by bringing charge onto a plate when some is already there (use W =QV) • Initially V = 0 • Average voltage during the charging process is V/2