Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley. Conservation of energy Work and Delta PE Electric potential energy Electric potential Chapter 21 Electric Potential Topics: Sample question: Shown is the electric potential measured on the surface of a patient. This potential is caused by electrical signals originating in the beating heart. Why does the potential have this pattern, and what do these measurements tell us about the heart’s condition? Slide 21-1

Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley. Chapter 21 Key Energy Equations Key Energy Equations from Physics 151 and Ch. 21 so far Definition of Work Where Work done by a conservative force (F g, F s, & F e ) Also work done by conservative force is path independent Conservation of Energy Equation (can ignore U g and U s unless they are relevant) Electric Energy – Special Cases (Similar equations for gravity) 2 Point Charges Charge in a Uniform E-field Note: in both cases of Electric Energy must assume where U e = 0 Slide Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley.

Chapter 21 Key Equations (3) Key Points about Electric Potential Electric Potential is the Electric Energy per Charge Electric Potential increases as you approach positive source charges and decreases as you approach negative source charges (source charges are the charges generating the electric field) A line where  V= 0 V is an equipotential line (The electric force does zero work on a test charge that moves on an equipotential line and  PE e = 0 J) Slide Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley.

Electric Potential and E-Field for Three Important Cases Slide For a point charge For very large charged plates, must use

Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley. The superposition principle and the V field due to multiple charges where Q 1, Q 2, Q 3, … are the source charges (including their signs) creating the field and r 1, r 2, r 3, … are the distances between the source charges and the location where we are determining the V field. So Electric Potentials (V) add just like Electric Potential Energies

Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley. Voting question Which of the following statements is incorrect? The E field is a vector. The electric potential is a scalar. The E field describes an interaction between charges. The electric potential energy describes an interaction between charges.

Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley. Quantitative Example Suppose that the heart's dipole charges −Q and +Q are separated by distance d. Write an expression for the V field due to both charges at point A, a distance d to the right of the +Q charge. 1. Simplify and diagram. 2. Represent mathematically.

Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley. Example Problem A.The potential at points a and b.The potential difference between a and b. B.The potential energy of a proton at a and b. C.The speed at point b of a proton that was moving to the right at point a with a speed of 4.0 x 10 5 m/s. D.The speed at point a of a proton that was moving to the left at point b with a speed of 4.0 x 10 5 m/s. For the situation shown in the figure, find Slide 21-22

Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley. Particles in a potential difference A positively charged object accelerates from regions of higher electric potential toward regions of lower potential (like an object falling to lower elevation in Earth's gravitational field). A negatively charged particle tends to do the opposite, accelerating from regions of lower potential toward regions of higher potential.

Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley. Slide Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley.

A Topographic Map Slide 21-12

Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley. Topographic Maps Slide Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley. 3. If a ball were placed at location D and another ball were placed at location C and both were released, which would have the greater acceleration? Which has the greater potential energy when released? Which will have a greater speed when at the bottom of the hill? 4. What factors does the speed at the bottom of the hill depend on? What factors does the acceleration of the ball depend on? 5. Is it possible to have a zero acceleration, but a non-zero height? Is it possible to have a zero height, but a non-zero acceleration? 1. Describe the region represented by this map. 2. Describe the directions a ball would roll if placed at positions A – D.

Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley. Equipotential surfaces: Representing the V field The lines represent surfaces of constant electric potential V, called equipotential surfaces. The surfaces are spheres (they look like circles on a two-dimensional page).

Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley. Equipotential surfaces and E field

Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley. Contour maps: An analogy for equipotential surfaces

Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley. Equipotential Maps (Contour Maps) Slide Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley. 5. At which point is the magnitude of the electric field the greatest? 6. Is it possible to have a zero electric field, but a non-zero electric potential? 7. Is it possible to have a zero electric potential, but a non-zero electric field? 1.Describe the charges that could create equipotential lines such as those shown above. Describe the forces a proton would feel at locations A and B. 3. Describe the forces an electron would feel at locations A and B Where could an electron be placed so that it would not move?

Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley. E-field lines and Equipotential lines Slide Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley. E-field Lines Go from + charges to - charges Perpendicular at surface of conductor or charged surface E-field in stronger where E-field lines are closer together More charge means more lines Equipotential Lines Parallel to conducting surface Perpendicular to E-field lines Near a charged object, that charges influence is greater, then blends as you to from one to the other E-field is stronger where Equipotential lines are closer together Spacing represents intervals of constant  V Higher potential as you approach a positive charge; lower potential as you approach a negative charge

Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley. Connecting Potential and Field Slide Note: Since the E-field everywhere inside a conductor is 0, V = constant

Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley. Polling Question The electric field A.is always perpendicular to an equipotential surface. B.is always tangent to an equipotential surface. C.always bisects an equipotential surface. D.makes an angle to an equipotential surface that depends on the amount of charge. Slide 21-12

Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley. Answer 4.The electric field A.is always perpendicular to an equipotential surface. B.is always tangent to an equipotential surface. C.always bisects an equipotential surface. D.makes an angle to an equipotential surface that depends on the amount of charge. Slide 21-13

Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley. 3D view Slide Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley.

Graphical Representations of Electric Potential Slide 21-13

Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley. Reading Quiz 3. The electric potential inside a parallel-plate capacitor A.is constant. B.increases linearly from the negative to the positive plate. C.decreases linearly from the negative to the positive plate. D.decreases inversely with distance from the negative plate. E.decreases inversely with the square of the distance from the negative plate. Slide 21-10

Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley. Answer 3. The electric potential inside a parallel-plate capacitor A.is constant. B.increases linearly from the negative to the positive plate. C.decreases linearly from the negative to the positive plate. D.decreases inversely with distance from the negative plate. E.decreases inversely with the square of the distance from the negative plate. Slide 21-11

Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley. The Potential Inside a Parallel-Plate Capacitor Slide 21-25

Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley. A.Rank the Electric Fields at points A, B, C, and D A.Rank the Electric Potentials at points A, B, C, and D Example Problem Source charges create the electric potential shown below. Slide 21-33

Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley. A.What is the potential at point A? At which point, A, B, or C, does the electric field have its largest magnitude? B.Is the magnitude of the electric field at A greater than, equal to, or less than at point D? Example Problem Source charges create the electric potential shown. C.What is the approximate magnitude of the electric field at point C? D.What is the approximate direction of the electric field at point C? Slide 21-33

Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley. Example Problem A proton is released from rest at point a. It then travels past point b. What is its speed at point b? Slide 21-23

Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley. Assembling a square of charges Slide Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley.

Analyzing a square of charges Slide Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley. Energy to Assemble W me =  E = U Ef - U Ei (U Ei = 0 J) U Ef = q 1 V + q 2 V + q 3 V + q 4 V V = V +V + V Energy to move (Move 2q from Corner to Center) W me =  E = U Ef - U Ei = q 2q V - q 2q V

Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley. Example Problem 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 21-26

Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley. What is Q 2 ? Example Problem Slide 21-35

Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley. Deriving a relation between the E field and ΔV We attach a small object with charge +q to the end of a very thin wooden stick and place the charged object and stick in the electric field produced by the plate. The only energy change is the system's electric potential energy, because the positively charged object moves farther away from the positively charged plate.

Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley. Deriving a relation between the E field andΔV Applying the generalized work-energy equation, we get: Equivalently, the component of the E field along the line connecting two points on the x-axis is the negative change of the V field divided by the distance between those two points: Ex = - Delta V / Delta x

Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley. Conceptual Exercise Can you think of locations relative to charge distributions where: 1.The V field at a particular location is zero but the E field is not? 2.The E field is zero but the V field is not zero?

Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley. Testing the relation between the E field and ΔV

Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley. Electric field of a charged conductor Free electrons in a conductor are quickly redistributed until equilibrium is reached, at which point the E field inside the conductor and parallel to its surface becomes zero.

Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley. Electric field outside a charged conductor

Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley. Grounding Grounding discharges an object made of conducting material by connecting it to Earth. Electrons will move between and within the spheres until the V field on the surfaces of and within both spheres achieves the same value.

Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley. Uncharged conductor in an electric field: Shielding The free electrons inside the object become redistributed due to electric forces, until the E field within the conducting object is reduced to zero.

Copyright © 2007, Pearson Education, Inc., Publishing as Pearson Addison-Wesley. Uncharged conductor in an electric field: Shielding The interior is protected from the external field—an effect called shielding.