Electricity and Magnetism. Electric Potential Energy and Voltage

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Scarlett York
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1 Electricity and Magnetism Electric Potential Energy and Voltage

2 Work and Potential Energy Recall from Mechanics that E mech = K + U is a conserved quantity for particles that interact via conservative forces and that for changes, E mech = K + U = 0. The change in potential energy is: U = U f U i = -W conservative force. If a particle moves a distance r while a constant force F is acting on it, then the work done is: W = F r = F r cos(θ), where θ is the angle between the force F and displacement r. There are three special cases: θ=0 0, θ=90 0, and θ= If the force is not constant, the work is: s f W = Fs d s = F d s s i f i 2

3 The Potential Energy in Two Uniform Fields The gravitational field g near the surface of the Earth is uniform. If a particle moves downward from y i to y f, the gravitational field will do a positive amount of work: Therefore: Wgrav = w r cos 0 = ( mg) y f yi = mg y U = U U = W = mg y grav f i grav Gravitational Potential Energy 3

cos 0 = ( mg) y f yi = mg y U = U U = W = mg y grav f i grav Gravitational Potential Energy Similarly, for displacements s in a

4 The Potential Energy in Two Uniform Fields The gravitational field g near the surface of the Earth is uniform. If a particle moves downward from y i to y f, the gravitational field will do a positive amount of work: Therefore: Wgrav = w r cos 0 = ( mg) y f yi = mg y U = U U = W = mg y grav f i grav Gravitational Potential Energy Similarly, for displacements s in a uniform electric field E, with s parallel to E: Welec = F r cos 0 = ( qe) s f si ( + 1) = qe s U = U U = W = qe s elec f i elec Electric Potential Energy 4

A positive charge gains energy as it moves away from the positive plate of a parallel plate capacitor, while a negative charge gains

5 Charges in an Electric Field One difference between a gravity field g and an electric field E is that a mass m interacting with g is always positive, while a charge q interacting with E may be either positive or negative. However, this is not a problem. A positive charge gains energy as it moves away from the positive plate of a parallel plate capacitor, while a negative charge gains energy as it moves away from the negative plate of the capacitor. In either case, the charge gains kinetic energy as its potential energy decreases. 5

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7 Example: Conservation of Energy inside a Capacitor A 2.0 cm x 2.0 cm parallel plate capacitor with a 2.0 mm gap is charged to ±1.0 nc. (Later in the year we will see that the electric field between the plates is 2.83 x 10 5 N/C) First a proton, and then an electron, are released at the midpoint of the capacitor. (a) What is each particle s change in potential energy ( U elec ) from its release to its collision with a plate? (b) What is each particle s kinetic energy as it reaches the plate? 7

8 Example: Conservation of Energy inside a Capacitor A 2.0 cm x 2.0 cm parallel plate capacitor with a 2.0 mm gap is charged to ±1.0 nc. (Later in the year we will see that the electric field between the plates is 2.83 x 10 5 N/C) First a proton, and then an electron, are released at the midpoint of the capacitor. (a) What is each particle s change in potential energy ( U elec ) from its release to its collision with a plate? (b) What is each particle s kinetic energy as it reaches the plate? 8

9 Conceptual Question 1 The electric field of a positively charged rod (end view shown) causes a negative particle to orbit the rod in a closed circular path, as shown. What is the sign of the work done on the charged particle by the electric field of the rod? (A) positive; (B) zero; (C) negative; (D) not enough information to tell. 9

Voltage In Chapter 23 we introduced the concept of an electric field E, which can be though of as a normalized force, i.e., E = F/q, the field E that would produce a force F on some test charge q.

Just like it is U that really matters and the actual values are arbitrary, it is changes in voltage V that we are going to be interested in.

10 Voltage In Chapter 23 we introduced the concept of an electric field E, which can be though of as a normalized force, i.e., E = F/q, the field E that would produce a force F on some test charge q. We can similarly define the voltage V as a charge-normalized potential energy, i.e., V=U elec /q, the voltage V that would give a test charge q an electric potential energy U elec because it is in the field of some other source charges. Just like it is U that really matters and the actual values are arbitrary, it is changes in voltage V that we are going to be interested in. We define the unit of voltage as the volt: 1 volt = 1 V = 1 J/C = 1 N m/c. 10

11 What Good is the Voltage? Like the electric field E, the voltage V is an abstract idea. It offers an advantage, however, because it is a scalar quantity while E is a vector, yet the two can be converted to each other. It is useful because: - The voltage depends only on the charges and their geometries. The voltage is the ability of the source charges to have an interaction if a charge q shows up. The voltage is present in all space, whether or not a charge is there to experience it. - If we know the voltage V throughout a region of space, we ll immediately know the potential energy U=qV of any charge q that enters that region. 11

12 Example: Moving Through a Voltage Difference A proton (q = 1.6 x C, m = 1.67 x kg) with a speed of v i = 2 x10 5 m/s enters a region of space where source charges have created a voltage. (a) What is the proton s final speed v f after it has moved through a voltage difference of V=100 V? (b) What is v f if the proton is replaced by an electron? 12

13 The Voltage Inside a Parallel Plate Capacitor Consider a parallel-plate capacitor with E = 500 N/C, to right Find the voltage difference (potential difference) between the two plates. 13

14 Graphical Representations of Electric Potential V x = = ( ) = 1 d d C V Es d x VC This linear relation can be represented as a graph, a set of equipotential surfaces, a contour plot, or a 3-D elevation graph. 14

15 Field Lines and Contour Lines Field lines and equipotential contour lines are the most widely used representations to simultaneously show the E field and the electric potential. The figure shows the field lines and equipotential contours for a parallel plate capacitor. Remember that both field lines and contours are virtual representations, not real objects, and that their spacing, etc, is a matter of choice. 15

16 Field Lines and Contour Lines For a constant electric field, if you know the voltage difference between two points, and how far apart the two points are, you can calculate the magnitude of the electric field from: E V = x To get the direction, just remember that the voltage decreases as you move in the direction that the electric field points. 16

17 Field Lines and Contour Lines If the electric field is not constant, you can use this method to estimate the strength of the electric field as long as x is small (the smaller x is, the closer E is to being constant in that interval). or more exactly E E V x lim V = = x 0 x dv dx We will use this method when we return to this topic and look at the parts that require calculus. 17

EField Java Field-Line Applet A special Java applet for plotting electric field lines, E-field gradients, and equipotential surfaces of any arrangement of point charges can be found at: http://www.

18 EField Java Field-Line Applet A special Java applet for plotting electric field lines, E-field gradients, and equipotential surfaces of any arrangement of point charges can be found at: The result looks like this: You must have a Java application available in order to run this applet. You are encouraged to use it to gain a better feeling for electric fields And equipotential lines. 18

19 Rules for Equipotentials 1. Equipotentials never intersect other equipotentials. (Why?) 2. The surface of any static conductor is an equipotential surface. The conductor volume is all at the same potential. 3. Field line cross equipotential surfaces at right angles. (Why?) 4. Close equipotentials indicate a strong electric field. The voltage V decreases in the direction in which the electric field E points, i.e., energetically downhill. 5. For any system with a net charge, the equipotential surfaces become spheres at large distances. 19

20 Conceptual Question 2 Which ranking of the voltages at points a-e is correct? (Ignore edge effects.) (a) V a >V b >V c >V d >V e (b) V a >V b =V c >V d =V e (c) V a =V b >V c >V d =V e (d) V a =V b =V c =V d =V e (e) V b >V a >V c >V e >V d 20

21 Conceptual Question 3 A proton and an electron are in a constant electric field created by oppositely charged plates. You release the proton from the positive side and the electron from the negative side. Which feels the larger electric force? 1) proton 2) electron 3) both feel the same force 4) neither there is no force 5) they feel the same magnitude force but opposite direction Electron electron proton - Proton + E

22 Conceptual Question 4 A proton and an electron are in a constant electric field created by oppositely charged plates. You release the proton from the positive side and the electron from the negative side. Which has the larger acceleration? 1) proton 2) electron 3) both feel the same acceleration 4) neither there is no acceleration 5) they feel the same magnitude acceleration but opposite direction Electron electron proton - Proton + E

23 Conceptual Question 5 A proton and an electron are in a constant electric field created by oppositely charged plates. You release the proton from the positive side and the electron from the negative side. When it strikes the opposite plate, which one has more KE? 1) proton 2) electron 3) both acquire the same KE 4) neither there is no change of KE 5) they both acquire the same KE but with opposite signs Electron electron proton - Proton + E

24 Conceptual Question 6 Which requires you to do the most work to move a positive charge from P to points 1, 2, 3 or 4? All points are the same distance from P. 1) P 1 2) P 2 3) P 3 4) P 4 5) all require the same amount of work P E 4

25 Conceptual Question 7 Which requires you to do zero work to move a positive charge from P to points 1, 2, 3 or 4? All points are the same distance from P. 1) P 1 2) P 2 3) P 3 4) P 4 5) all require the same amount of work P E 4

26 The voltage of a point charge (letting the voltage be zero infinitely away from the charges) is given by: V The Voltage of a Point Charge kq 1 q = = Example: q = 1 nc, r = 1 cm; r 4πε r We will show that this equation is correct using calculus later in the year. For now we are just interested in using it. 0 V = kq r ( C) ( Nm /C ) ( m) = = 900 V You would use the given equation to find the voltage at this point due to the source charge q. 26

27 Conceptual Question 8 Which ranking of the potential differences is correct? (a) V 12 > V 23 > V 13 (b) V 12 < V 23 < V 31 (c) V 12 < V 23 = V 13 (d) V 12 = V 23 > V 13 (e) V 12 = V 23 = V 13 27

28 Visualizing the Voltage of a Point Charge The potential of a point charge can be represented as a graph, a set of equipotential surfaces, a contour map, or a 3-D elevation graph. Usually it is represented by a graph or a contour map, possibly with field lines. + 28

29 Conceptual Question 9 Which two points are at the same potential (voltage)? 1) A and C 2) B and E 3) B and D 4) C and E 5) no pair A C B E Q D

30 The Voltage of Many Charges The principle of superposition allows us to calculate the voltages created by many point charges and then add the up. Since the voltage V is a scalar quantity, the superposition of potentials is simpler than the superposition of fields. V = i kq r i i 30

31 Example: The Voltage of Two Charges What is the voltage at point p? Let V = 0 at r = p 31

Electric Fields Part 1: Coulomb s Law

Electric Fields Part 1: Coulomb s Law F F Last modified: 07/02/2018 Contents Links Electric Charge & Coulomb s Law Electric Charge Coulomb s Law Example 1: Coulomb s Law Electric Field Electric Field Vector