Physics 122B Electricity and Magnetism

1. Physics 122B Electricity and Magnetism Lecture 11 (Knight: 29.5 to 29.7) Electric Potential, Equipotential Surfaces and E = -r V April 20, 2007 Martin Savage

2. Lecture 11 Announcements • Lecture HW Assignment #4 is due at 10 PM, next Wednesday. • Uncollected Exam 1 papers may be obtained from Helen Gribble in room C136 PAB. • Requests for regrades of Exam 1 should be written on a separate sheet (see Syllabus) and taken to Helen Gribble in room C136 PAB. They will be accepted until noon next Wednesday. Physics 122B - Lecture 11

3. Midterm 1 Stats Physics 122B - Lecture 11

4. The Electric Potential In Chapter 25 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 electric potentialV as a charge-normalized potential energy, i.e., V=Uelec/q, the electric potential V that would give a test charge q an electric potential energy Uelecbecause it is in the field of some other source charges. We define the unit of electric potential as the volt: 1 volt = 1 V =1 J/C = 1 Nm/C. Other units are: kV=103 V, mV=10-3 V, and mV=10-6 V. Example: A D-cell battery has a potential of 1.5 V between its terminals. Physics 122B - Lecture 11

5. What Good isthe Electric Potential? • Like the electric field E, the electric potential 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 electric potential V depends only on the charges and their geometries. The electric potential is the “ability” of the source charges to have an interaction if a charge q shows up. The potential is present in all space, whether or not a charge is there to experience it. • If we know the electric potential V throughout a region of space, we’ll immediately know the potential energy U = qV of any charge q that enters that region. Gaining Potential Energy Losing Kinetic Energy Losing Potential Energy Gaining Kinetic Energy Physics 122B - Lecture 11

6. Example: Moving Through a Potential Difference A proton with a speed of vi = 2.0 x 105 m/s enters a region of space where source charges have created an electric potential. What is the proton’s speed after it has moved through a potential difference of DV = 100 V? What is vf if the proton is replaced by an electron? Physics 122B - Lecture 11

7. The Electric Potential Insidea Parallel Plate Capacitor Consider a parallel-plate capacitor with (with U0=0) Physics 122B - Lecture 11

8. Graphical Representationsof Electric Potential Distance from + plate This linear relation can be represented as a graph, a set of equipotential surfaces, a contour plot, or a 3-D elevation graph. Physics 122B - Lecture 11

9. 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 for both the field lines and contours , their spacing, etc, is a matter of choice. Physics 122B - Lecture 11

10. Rules for Equipotentials • Equipotentials never intersectother equipotentials. (Why?) • The surface of any staticconductor is an equipotentialsurface. The conductor volumeis all at the same potential. • Field line cross equipotentialsurfaces at right angles. (Why?) • Dense equipotentials indicate astrong electric field. The potential V decreases in the direction in which the electric field E points, i.e., energetically “downhill” for a + charge • For any system with a net charge, the equipotential surfaces become spheres at large distances. Physics 122B - Lecture 11

11. Batteries and Capacitors How can we arrange for a capacitor to have a surface charge density of precisely h=4.42 x 10-9 C/m2, as in the previous example? In Chapter 28 we introduced batteries, which are chemical sources of constant electric potential difference. By choosing a battery that supplies a potential differenceE = hd/e0, and by arranging the plate-separation, we can place any desired charge density hon capacitor plates. h =Q/A= 4.42x10-9 C/m2 Physics 122B - Lecture 11

12. Example: A Proton in a Capacitor • A parallel plate capacitor is made of two 2.0 cm diameter disks spaced 2.0 mm apart. It is charged to a potential difference of 500 V. • What is the E field in the gap? • How much charge is on each plate? • A proton is shot through a small hole in the negative plate with a speed of v=2.0x105 m/s. Does it reach the other side? If not where is the turning point? Physics 122B - Lecture 11

13. Choice of the V=0 Point In the previous example we assumed that the negative plate of the capacitor was the V=0 point. However, we could just as well have placed the V=0 point at the right plate, or half way between the two plates, since only the potential difference DVCmatters in calculations. 0 V Physics 122B - Lecture 11

14. Question Which ranking of the potentials at points a-e is correct? (Ignore edge effects.) • Va>Vb>Vc>Vd>Ve • Va>Vb=Vc>Vd=Ve • Va=Vb>Vc>Vd=Ve • Va=Vb=Vc=Vd=Ve • Vb>Va>Vc>Ve>Vd Physics 122B - Lecture 11

15. The Electric Potentialof a Point Charge Example: q = 1 nC, r = 1 cm; Then divide Uq’+q by q’. Physics 122B - Lecture 11

16. Visualizing the Potentialof 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. + Spherical Shells Physics 122B - Lecture 11

17. Question Which ranking of the potentials differences is correct? • DV12>DV23>DV13 • DV12<DV23<DV31 • DV12<DV23=DV13 • DV12=DV23>DV13 • DV12=DV23=DV13 Physics 122B - Lecture 11

18. Q R The Electric Potentialof a Charged Sphere (same as for point charge Q at center) (potential at surface of sphere) Physics 122B - Lecture 11

19. Example: A Proton anda Charged Sphere • A proton is released fromrest on the surface ofa 1.0 cm diameter spherethat has been chargedto +1000 V. • What is the charge ofthe sphere? • What is the proton’s speed after it travels 1.0 cm from the sphere? Physics 122B - Lecture 11

20. The Electric Potentialof Many Charges The principle of superposition allows us to calculate the potentials created by many point charges and then add the up. Since the potential V is a scalar quantity, the superposition of potentials is simpler than the superposition of fields. Physics 122B - Lecture 11

21. Example: The Potentialof Two Charges p What is the potential at point p? Note that: 1/4pe0 = 9.0 x 109 Nm2/C2 = 9.0 x 109 Vm/C,which, for problems like this, are more convenient units. Physics 122B - Lecture 11

22. Example: The Potentialof a Ring of Charge Find the potential of a thin uniformly charged ring of radius R and charge Q at point P on the z axis? Physics 122B - Lecture 11

23. Example: The Potentialof a Disk of Charge Find the potential of a uniformly charged disk of radius R and charge Q at point P on the z axis? P Physics 122B - Lecture 11

24. Potential of a Disk of Charge Physics 122B - Lecture 11

25. Example: The Potential of a Dime • A dime (diameter 17.5 mm) is given a charge of Q=+5.0 nC. • What is the potential of the dime at its surface? • What is the potential energy Ue of an electron 1.0 cm above the dime (on axis)? + + + + + + + + + + + + Physics 122B - Lecture 11

26. End of Lecture 11 • Before the next lecture, read Knight, Chapters 30.1 through 30.4. • Lecture HW Assignment #4 is due at 10 PM, next Wednesday. • Uncollected Exam 1 papers may be obtained from Helen Gribble in room C136 PAB. • Requests for regrades of Exam 1 should be written on a separate sheet (see Syllabus) and taken to Helen Gribble in room C136 PAB. They will be accepted until noon next Wednesday. Physics 122B - Lecture 11