Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 1 Chapter 17: Electric Potential Electric Potential Energy Electric Potential How are the E-field and Electric Potential related? Motion of Point Charges in an E-field Capacitors Dielectrics

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 2 §17.1 Electric Potential Energy Electric potential energy (U e ) is energy stored in the electric field. U e depends only on the location, not upon the path taken to get there (conservative force). U e = 0 at some reference point. For two point particles take U e = 0 at r = . For the electric force

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 3 Example: A proton and an electron, initially separated by a distance r, are brought closer together. How does the potential energy of this system of charges charge? Bringing the charges closer together decreases r:. For these two charges This is like a mass falling near the surface of the Earth; positive work is done by the field.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 4 When q 1 and q 2 have the same algebraic sign then  U e > 0. This means that work must be done by an external agent to bring the charges closer together. Example continued How will the electric potential energy change if both particles have positive (or negative) charges?

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 5 What is the potential energy of a system (arrangement) of point charges? To calculate: Begin by placing the first charge at a place in space far from any other charges. No work is required to do this. Next, bring in the remaining charges one at a time until the desired configuration is finished.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 6 Example: What is the potential energy of three point charges arranged as a right triangle? (See text Example 17.2) Are these the same?

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 7 §17.2 Electric Potential Electric potential is the electric potential energy per unit charge. Electric potential (or just potential) is a measurable scalar quantity. Its unit is the volt (1 V = 1 J/C).

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 8 For a point charge of charge Q: When a charge q moves through a potential difference of  V, its potential energy change is  U e = q  V.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 9 Q b a c e d g f Example: A charge Q = +1 nC is placed somewhere in space far from other charges. Take r a = r b = r c = r d = 1.0 m and r e = r f = r g = 2.0 m.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 10 (a) Compare the potential at points d and g. Example continued: Since Q>0 the potential at point d is greater than at point g, it is closer to the charge Q. (b) Compare the potential at points a and b. The potential at point a is the same as at point b; both are at the same distance from the charge Q.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 11 (c) Place a charge of nC at point e. What will the change in potential (  V) be if this charge is moved to point a?  V = V f – V i = V a -V e = +4.5 Volts Example continued:

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 12  U e =q  V = (+0.50 nC)(+4.5 Volts)= +2.3 nJ (d) What is the change in potential energy (  U) of the nC charge ? Example continued:

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 13 (e) How would the results of the previous questions change if instead of a +1.0 nC charge there is a -1.0 nC charge in its place? Example continued: (a)The potential at point d is less than the potential at point g. (b) Unchanged (c) -4.5 V (d) -2.3 nJ

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 14 §17.3 The Relationship between E and V Q b a c e d g f +9 V +4.5 V The circles are called equipotentials (surfaces of equal potential).

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 15 The electric field will point in the direction of maximum potential decrease and will also be perpendicular to the equipotential surfaces. Q b a c e d g f +9 V +4.5 V

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 16 Equipotentials and field lines for a dipole.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 17 Uniform E-field E Equipotential surfaces V1V1 V2V2 V3V3 V4V4 Where d is the distance over which  V occurs.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 18 If the electric field inside a conductor is zero, what is the value of the potential? If E=0, then  V=0. The potential is constant! What is the value of V inside the conductor? It will be the value of V on the surface of the conductor.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 19 §17.4 Moving Charges When only electric forces act on a charge, its total mechanical energy will be conserved.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 20 Example (text problem 17.31): Point P is at a potential of kV and point S is at a potential of kV. The space between these points is evacuated. When a charge of +2e moves from P to S, by how much does its kinetic energy change?

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 21 Example (text problem 17.32): An electron is accelerated from rest through a potential difference. If the electron reaches a speed of 7.26  10 6 m/s, what is the potential difference? 0 Note: the electron moves from low V to high V.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 22 §17.5 Capacitors A capacitor is a device that stores electric potential energy by storing separated positive and negative charges. Work must be done to separate the charges. Parallel plate capacitor

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 23 Written as an equality: Q = C  V, where the proportionality constant C is called the capacitance. For a parallel plate capacitor:

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 24 Note: C depends only on constants and geometrical factors. The unit of capacitance is the farad (F). 1 F = 1 C 2 /J = 1 C/V What is the capacitance for a parallel plate capacitor?

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 25 Example (text problem 17.42): A parallel plate capacitor has a capacitance of 1.20 nF. There is a charge of magnitude  C on each plate. (a) What is the potential difference between the plates?

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 26 (b) If the plate separation is doubled, while the charge is kept constant, what will happen to the potential difference? If d is doubled so is the potential difference. Example continued:

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 27 Example (text problem 17.86): A parallel plate capacitor has a charge of  C on each plate with a potential difference of 240 volts. The parallel plates are separated by 0.40 mm of air. (a) What is the capacitance of this capacitor?

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 28 (b) What is the area of a single plate? Example continued:

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 29 §17.6 Dielectrics As more and more charge is placed on capacitor plates, there will come a point when the E-field becomes strong enough to begin to break down the material (medium) between the capacitor plates.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 30 To increase the capacitance, a dielectric can be placed between the capacitor plates. and  is the dielectric constant.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 31 Example (text problem 17.55): A capacitor can be made from two sheets of aluminum foil separated by a sheet of waxed paper. If the sheets of aluminum are 0.3 m by 0.4 m and the waxed paper, of slightly larger dimensions, is of thickness mm and has  = 2.5, what is the capacitance of this capacitor?

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 32 §17.7 Energy Stored in a Capacitor A capacitor will store energy equivalent to the amount of work that it takes to separate the charges.

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 33 These are found by using Q=C  V and the first relationship. The energy stored in the electric field between the plates is: }

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 34 Example (text problem 17.63): A parallel plate capacitor is composed of two square plates, 10.0 cm on a side, separated by an air gap of 0.75 mm. (a) What is the charge on this capacitor when the potential difference is 150 volts? (b) What energy is stored in this capacitor?

Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 35 Summary Electric Potential Energy Electric Potential The Relationship Between E and V Motion of Point Charges (conservation of energy) Parallel Plate Capacitors (capacitance, dielectrics, energy storage)