Chapter 21 Electric Potential
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1 Chapter 21 Electric Potential Chapter Goal: To calculate and use the electric potential and electric potential energy. Slide 21-1
2 Chapter 21 Preview Looking Ahead Text: p. 665 Slide 21-2
3 Review of Potential Energy Consider a particle being acted on by a conservative force F. If the displacement of the particle is a distance d to the right, the amount of work done on the particle by this force is W = F k d = Fdcos Recall that the change in potential energy associated with a conservative force is related to the work done by the force: U = W cons U = F cons d cos Slide 21-3
4 Gravitational Potential Energy The force of gravity is If the book is lifted a distance h, the change in potential energy is U g = ~F g = mgŷ W g = mgh An alternate way to think about the change in potential energy is to consider how much work you would have to do in lifting the book. If the book is raised such that its kinetic energy doesn t change, the net work must be zero. Thus the work you do in lifting the book is negative the work done by the force of gravity. W ext = W g =+ U g Slide 21-4
5 Electric Potential Energy and Electric Potential The hand does positive work in compressing (or stretching) a spring Since the force isn t constant, we have to sum up the the individual amount of work done for small displacements (calculus). X U s = F sp (x i ) x i i F sp = kx U s = 1 2 kx2 Slide 21-5
6 Electric Potential Energy and Electric Potential A charge q is repelled by stationary source charges. A hand must push on the charge q in order to move it closer to the source charges. The energy is electric potential energy U elec. Slide 21-6
7 Relationship between Electric Potential Energy and E Fields U = For electric potential energy, the conservative force is due to the electric field. Thus, for displacements in a uniform electric field, U elec = F cons d cos ~F elec = q ~ E If the electric field were not uniform, U elec = qe d cos q X i (for uniform force) E(r i ) r i cos i Slide 21-7
8 Clicker Question Consider uniform electric field (say inside a parallel capacitor) A B!!!!!! C If a proton is taken from location A to location B, how does its potential energy change? 1. it decreases 2. it increases 3. it doesn t change Slide 21-8
9 Clicker Question Consider uniform electric field (say inside a parallel capacitor) A B!!!!!! C If a proton is taken from location B to location C, how does its potential energy change? 1. it decreases 2. it increases 3. it doesn t change Slide 21-9
10 Clicker Question A B!!!!!! C The potential difference of a proton that goes from A to C is in magnitude to that of a proton going from B to C. 1. greater 2. smaller 3. equal Slide 21-10
11 Whiteboard Example A B d!!!!!! C y = 0 A proton is released from rest just below the top (positive) plate of an parallel plate capacitor with an electric field strength E = 100 N/C. If the distance between the plates is d = 3 mm, how fast is it moving when it hits the bottom (negative) plate? Slide 21-11
12 Electric Potential Energy of Two Point Charges Now consider a charge Q at origin and another charge q that is initially very far away. The work required to bring charge q from infinity to r is (using calculus) U elec = q X i E(r i ) r i cos i U elec = q X i kq x 2 i ( x i )= kqq r If we pick infinity as the zero point for potential energy, then U elec (r) = kqq r Slide 21-12
13 Example: Rutherford Scattering Rutherford scattering. A helium nucleus of mass 4 m is emitted with an initial speed of v 0 = 4.9 x 10 5 m/s towards a gold nucleus of charge q 2 = 79 e. What is the minimum distance between the two particles (assume the gold nucleus doesn t move)? Slide 21-13
14 Electric Potential (Voltage) Notice that the electric potential energy of a particle depends on both on the electric field produced by the sources and the charge of the particle. In all cases the potential energy is proportional to the particle s charge. We define the electric potential (or Voltage), V, such that the electric potential energy of a charge q is given by Slide 21-14
15 Electric Potential (Voltage) vs. Potential Energy Potential energy deals with the energy of a particle. Like the electric force, it only makes sense to talk about potential energy of a particle. Voltage is a property of space. Like the electric field, there can be a voltage at some location even if no particle exists at that location. The electric field tells us how a source will exert a force on q; the electric potential tells us how a charge q will have potential energy. Slide 21-15
16 Clicker Question In a parallel plate capacitor, the electric field is uniform and is directed from the positive plate to the negative plate. An electron goes from location A to location C. Which statement is true? A) The electron goes from a high voltage to a lower voltage. B) The electron goes from a low voltage to a higher voltage. C) The voltage is the same at both locations. Slide 21-16
17 Clicker question The figure shows three straight paths AB of the same length, each in a different electric field. Which one of the three has the largest magnitude of a voltage difference between the two points? A. (a) B. (b) C. (c) Slide 21-17
18 Electric Potential Slide 21-18
19 Electric Potential and Conservation of Energy The conservation of energy equation for a charged particle is K f + (U elec ) f = K i + (U elec ) i In terms of electric potential V the equation is Slide 21-19
20 Whiteboard Example A parallel-plate capacitor is held at a potential difference of 250 V. A proton is fired toward a small hole in the negative plate with a speed of 3.0 x 10 5 m/s. What is its speed when it emerges through the hole in the positive plate? (Hint: The electric potential outside of a parallel-plate capacitor is zero). Slide 21-20
21 Voltage of a point charge Recall the potential energy of two point charges: U(r) =k q 0q r Thus the voltage a distance r from the charge q is given by V (r) =U(r)/q 0 = kq r Slide 21-21
22 Clicker Question A conducting sphere of radius R has a net charge Q (on its surface). What is the voltage at the center of the sphere? 1. V = kq/r 2. V > kq/r 3. 0 < V < kq/r 4. V=0 Slide 21-22
23 QuickCheck What is the electric potential at the surface of the sphere? A. 15 V B. 30 V C. 60 V D. 90 V E. 120 V R = 5 cm 15 cm V = 30 V Slide 21-23
24 Example: Maximum voltage of a Van de Graaff generator. Molecules in air get ionized for electric fields greater than roughly E max = 3 x 10 6 V/m. What is the maximum voltage of a charged sphere of radius R=0.2 m? Slide 21-24
25 QuickCheck An electron follows the trajectory shown from point 1 to point 2. At point 2, A. v 2 > v 1 B. v 2 = v 1 C. v 2 < v 1 D. Not enough information to compare the speeds at these points Slide 21-25
26 The Electric Potential of Many Charges Suppose there are many source charges, q 1, q 2 The electric potential V at a point in space is the sum of the potentials due to each charge. r i is the distance from charge q i to the point in space where the potential is being calculated. Slide 21-26
27 Clicker Question Two identical positive charges of charge Q are a distance d apart. What is the voltage at the midway point between the charges? a) k Q/d b) 2 k Q/d c) 4 k Q/d d) 8 k Q/d e) 0 Slide 21-27
28 Question Hollywood Square Four point charges are arranged at the corners of a square. Find the electric field E and the potential V at the center of the square. 1) E = 0 V = 0 2) E = 0 V 0 3) E 0 V 0 4) E 0 V = 0 5) E = V regardless of the value -Q +Q -Q +Q Slide 21-28
29 QuickCheck Four charges lie on the corners of a square with sides of length a. What is the electric potential at the center of the square? a + q + q a q q Slide 21-29
30 Equipotential Diagrams An equipotential is a surface on which the potential (voltage) is constant. In two-dimensional drawings, equipotential curves are similar to contours of an elevation map. Slide 21-30
31 Clicker Question Slide 21-31
32 QuickCheck 21.7 A proton is released from rest at the dot. Afterward, the proton A. Remains at the dot. B. Moves upward with steady speed. C. Moves upward with an increasing speed. D. Moves downward with a steady speed. E. Moves downward with an increasing speed. Slide 21-32
33 The Electric Potential Inside a Parallel-Plate Capacitor Below are graphical representations of the electric potential inside a capacitor. Slide 21-33
34 Clicker question The figure shows cross sections through two equipotential surfaces. In both diagrams the potential difference between adjacent equipotentials is the same. Which of these two could represent the field of a point charge? A. (a) B. (b) C. neither (a) nor (b) Slide 21-34
35 Example: Electric Dipole The potential of an electric dipole is the sum of the potentials of the positive and negative charges. Slide 21-35
36 Conductors There s no electric field inside a conductor in electrostatic equilibrium. At the surface there s no parallel component of the electric field. Therefore in electrostatic equilibrium, the entire conductor is at the same potential (the surface is an equipotential). Slide 21-36
37 The Electrocardiogram A measurement of the electric potential of the heart is an invaluable diagnostic tool. The potential difference in a patient is measured between several pairs of electrodes. A chart of the potential differences is the electrocardiogram, also called an ECG or an EKG. Slide 21-37
38 Connecting Potential and Field Recall the relationship between the electric field and the change in voltage. X V = i For a very small displacement Δr in the same direction as the electric field (perpendicular to an equipotential surface), V = E r E(r i ) r i cos i Thus the electric field is given by E = V r Slide 21-38
39 QuickCheck Which set of equipotential surfaces matches this electric field? E r A. 0 V 50 V 0 V 50 V B. 0 V 50 V 50 V 0 V C. D. E. 50 V 0 V Slide 21-39
40 Clicker Question For which region is the magnitude of the electric field the highest? Slide 21-40
41 Example Problem What are the magnitude and direction of the electric field at the dot? Slide 21-41
42 Capacitors A capacitor is a pair of conductors (plates), insulated from each other, and used to store charge and energy. For a charged capacitor, one conductor is positively charged and the other is negatively charged (net charge is always zero). The work used in separating charge is stored as electrostatic energy in the capacitor. Slide 21-42
43 Capacitance and Capacitors The potential difference between the electrodes is directly proportional to the amount they are charged. Q = CV V refers to magnitude of the voltage difference between conductors The constant of proportionality, C, is the capacitance. The SI unit of capacitance is the farad. Slide 21-43
44 Charging a Capacitor Slide 21-44
45 QuickCheck What is the capacitance of these two electrodes? A. 8 nf B. 4 nf C. 2 nf D. 1 nf E. Some other value Slide 21-45
46 Parallel Plate Capacitor What is the capacitance of a parallel plate capacitor of area A and separation d? Determine E: E = 1 0 Determine V in terms of Q V = Ed = 1 d = Qd A Determine C: C = Q V = 0 Capacitance is always independent of voltage and charge! A d Slide 21-46
47 QuickCheck A capacitor is attached to a battery. The plates are then pulled apart so that the distance between them is larger. After the plates are pulled apart, A. The charge increases and the electric field decreases. B. The charge decreases and the electric field increases. C. Both the charge and the field increase. D. Both the charge and the field decrease. E. The charge and the field remain constant. Slide 21-47
48 Example Problem A parallel-plate capacitor is constructed of two square plates, 1 m on each side, separated by a 1.0 mm gap. What is the capacitance of this capacitor? If it were charged to 100 V, how much charge would be on the capacitor? Slide 21-48
49 Practical capacitors Capacitors have capacitances ranging from picofarads (pf; F) to several farads. Most use a dielectric material between their plates. The dielectric constant, apple, is a property of the dielectric material that gives the reduction of electric field. E = E 0 /apple Slide 21-49
50 C = Q V = Q V 0 /apple = applec 0 Slide 21-50
51 Energy stored in a capacitor Charging a capacitor involves transferring charge between the initially neutral plates. The work dw involved in moving charge dq is dw=δv(q) dq The voltage difference increases as the capacitor charges up. The total work done in charging the capacitor to a charge Q is Slide 21-51
52 Energy and Capacitors The potential energy U C stored in a charged capacitor is Since Q = C ΔV C, this can be written as This is the electrostatic energy stored in the capacitor. Slide 21-52
53 CT 29.C4 A parallel plate capacitor is charged (the plates are isolated so Q cannot change.) The plates are then pulled apart so that the plate separation d increases. The total electrostatic energy stored in the capacitor. d Q -Q A:increases B:decreases C: stays same E University of Colorado, Boulder
54 QuickCheck A capacitor charged to 1.5 V stores 2.0 mj of energy. If the capacitor is charged to 3.0 V, it will store A. 1.0 mj B. 2.0 mj C. 4.0 mj D. 6.0 mj E. 8.0 mj Slide 21-54
55 Energy and Capacitors A capacitor can discharge and release its stored energy very quickly. A medical application of this ability to rapidly deliver energy is the defibrillator. Fibrillation is the state in which the heart muscles twitch and cannot pump blood. A defibrillator is a large capacitor that can store up to 360 J of energy and release it in 2 milliseconds. The large shock can sometimes stop fibrillation. Slide 21-55
56 Energy in the electric field What exactly is electrostatic potential energy? How is it a form of energy? Postulate: the electrostatic energy associated with a charge distribution is due to the electric field Electric fields are a form of energy! Energy density: u E = 1 2 ε 0 E 2 U = 1 2 CV 2 = u E Ad (C = 0 A/d) (V 2 = E 2 d 2 ) Slide 21-56
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