PHY 184

1. PHY 184 Spring 2007 Lecture 9 Title: The Electric Potential 184 Lecture 9

2. Announcements • Homework Set 2 done, Set 3 ongoing and Set 4 will open on Thursday • Helproom hours of the TAs are listed on the syllabus in LON-CAPA • Honors Option students will provide help in the in the SLC starting this week. • Remember Clicker’s Law… Up to 5% (but not more!) 184 Lecture 9

3. Review - Potential Energy • When an electrostatic force acts between charged particles, assign an electric potential energy, U. • The difference in U of the system in two different states, initial i and final f, is • Reference point: Choose U=0 at infinity. • If the system is changed from initial state i to the final state f, the electrostatic force does work, W • Potential energy is a scalar. 184 Lecture 9

4. Review - Work • Work done by an electric field Q is the angle between electric field and displacement (1) Positive W  U decreases (2) Negative W  U increases 184 Lecture 9

5. Clicker Question • In the figure, a proton moves from point i to point f in a uniform electric field directed as shown. Does the electric field do positive, negative or no work on the proton? A: positive B: negative C: no work is done on the proton 184 Lecture 9

6. Clicker Question • In the figure, a proton moves from point i to point f in a uniform electric field directed as shown. Does the electric field do positive, negative or no work on the proton? B: negative 184 Lecture 9

7. Electric Potential V • The electric potential, V, is defined as the electric potential energy, U, per unit charge • The electric potential is a characteristic of the electric field, regardless of whether a charged object has been placed in that field. (because U  q) • The electric potential energy is an energy of a charged object in an external electric field (or more precisely, an energy of the system consisting of the charged object and the external field). 184 Lecture 9

8. Electric Potential Difference DV • The electric potential difference between an initial point i and final point f can be expressed in terms of the electric potential energy of q at each point • Hence we can relate the change in electric potential to the work done by the electric field on the charge 184 Lecture 9

9. Electric Potential Difference (2) • Taking the electric potential energy to be zero at infinity we have where We, is the work done by the electric field on the charge as it is brought in from infinity. • The electric potential can be positive, negative, or zero, but it has no direction. (i.e., scalar not vector) • The SI unit for electric potential is joules/coulomb, i.e., volt. Explain: i =  , f = x, so that DV = V(x) - 0 184 Lecture 9

10. The Volt • The commonly encountered unit joules/coulomb is called the volt, abbreviated V, after the Italian physicist Alessandro Volta (1745 - 1827) • With this definition of the volt, we can express the units of the electric field as • For the remainder of our studies, we will use the unit V/m for the electric field. 184 Lecture 9

11. Example - Energy Gain of a Proton - + • A proton is placed between two parallel conducting plates in a vacuum as shown. The potential difference between the two plates is 450 V. The proton is released from rest close to the positive plate. • What is the kinetic energy of the proton when it reaches the negative plate? The potential difference between the two plates is 450 V. = V(+)-V(-) The change in potential energy of the proton is DU, and DV = DU / q (by definition of V), so DU = q DV = e[V(-)-V(+)] = -450 eV 184 Lecture 9

12. Example - Energy Gain of a Proton (2) • Because the acceleration of a charged particle across a potential difference is often used in nuclear and high energy physics, the energy unit electron-volt (eV) is common. • An eV is the energy gained by a charge e that accelerates across an electric potential of 1 volt • The proton in this example would gain kinetic energy of 450 eV = 0.450 keV. Conservation of energy DK = - DU = + 450 eV initial final Because the proton started at rest, K = 1.6x10-19 C x 450 V = 7.2x10-17 J 184 Lecture 9

13. The Van de Graaff Generator • A Van de Graaff generator is a device that creates high electric potential. • The Van de Graaff generator was invented by Robert J. Van de Graaff, an American physicist (1901 - 1967). • Van de Graaff generators can produce electric potentials up to many 10s of millions of volts. • Van de Graaff generators can be used to produce particle accelerators. • We have been using a Van de Graaff generator in lecture demonstrations and we will continue to use it. 184 Lecture 9

14. The Van de Graaff Generator (2) • The Van de Graaff generator works by applying a positive charge to a non-conducting moving belt using a corona discharge. • The moving belt driven by an electric motor carries the charge up into a hollow metal sphere where the charge is taken from the belt by a pointed contact connected to the metal sphere. • The charge that builds up on the metal sphere distributes itself uniformly around the outside of the sphere. • For this particular Van de Graaff generator, a voltage limiter is used to keep the Van de Graaff generator from producing sparks larger than desired. 184 Lecture 9

15. The Tandem Van de Graaff Accelerator C+6 C-1 Terminal at +10MV • One use of a Van de Graaff generator is to accelerate particles for condensed matter and nuclear physics studies. • Clever design is the tandem Van de Graaff accelerator. • A large positive electric potential is created by a huge Van de Graaff generator. • Negatively charged C ions get accelerated towards the +10 MV terminal (they gain kinetic energy). Stripper foil strips electrons from C Electrons are stripped from the C and the now positively charged C ions are repelled by the positively charged terminal and gain more kinetic energy. 184 Lecture 9

16. Example - Energy of Tandem Accelerator • Suppose we have a tandem Van de Graaff accelerator that has a terminal voltage of 10 MV (10 million volts). We want to accelerate 12C nuclei using this accelerator. • What is the highest energy we can attain for carbon nuclei? • What is the highest speed we can attain for carbon nuclei? • There are two stages to the acceleration • The carbon ion with a -1e charge gains energyaccelerating toward the terminal • The stripped carbon ion with a +6e charge gainsenergy accelerating away from the terminal 15 MV Tandem Van de Graaff at Brookhaven 184 Lecture 9

17. Example - Energy of Tandem Accelerator (2) 184 Lecture 9

18. Equipotential Surfaces and Lines Equipotential surface from eight point chargesfixed at the corners of a cube • When an electric field is present, the electric potential has a given value everywhere in space. V(x) = potential function • Points close together that have the same electric potential form an equipotential surface. i.e, V(x) = constant value • If a charged particle moves on an equipotential surface, no work is done. • Equipotential surfaces exist in threedimensions. • We will often take advantageof symmetries in the electric potentialand represent the equipotential surfacesas equipotential lines in a plane. 184 Lecture 9

19. General Considerations if d  E • If a charged particle moves perpendicular to electric field lines, no work is done. • If the work done by the electric field is zero, then the electric potential must be constant • Thus equipotential surfaces and lines must always be perpendicular to the electric field lines. 184 Lecture 9

20. Electric field lines and equipotential surfaces 184 Lecture 9

21. Constant Electric Field • Electric field lines: straight lines parallel to E • Equipotential surfaces (3D): planes perp to E • Equipotential lines (2D): straight lines perp to E 184 Lecture 9

22. Electric Field from a Single Point Charge • Electric field lines: radial lines emanating from the point charge. • Equipotential surfaces (3D): concentric spheres • Equipotential lines (2D): concentric circles 184 Lecture 9

23. Electric Field from Two Oppositely Charged Point Charges • The electric field lines from two oppositely charge point charges are a little more complicated. • The electric field lines originate on the positive charge and terminate on the negative charge. • The equipotential lines are always perpendicular to the electric field lines. • The red lines represent positiveelectric potential. • The blue lines represent negativeelectric potential. • Close to each charge, the equipotentiallines resemble those from a pointcharge. 184 Lecture 9

24. ELECTRIC DIPOLE 184 Lecture 9

25. Electric Field from Two Identical Point Charges • The electric field lines from two identical point charges are also complicated. • The electric field lines originate on the positive charge and terminate at infinity. • Again, the equipotential linesare always perpendicular tothe electric field lines. • There are only positivepotentials. • Close to each charge, theequipotential lines resemblethose from a point charge. 184 Lecture 9

26. TWO POSITIVE CHARGES 184 Lecture 9

27. Calculating the Potential from the Field • To calculate the electric potential from the electric field we start with the definition of the work dW done on a particle with charge q by a force F over a displacement ds • In this case the force is provided by the electric fieldF = qE • Integrating the work done by the electric force on the particle as it moves in the electric field from some initial point i to some final point f we obtain 184 Lecture 9

28. Calculating the Potential from the Field (2) • Remembering the relation between the change in electric potential and the work done … • …we find • Taking the convention that the electric potential is zero at infinity we can express the electric potential in terms of the electric field as ( i = , f = x) 184 Lecture 9

29. Example - Charge moves in E field • Given the uniform electric field E, find the potential difference Vf-Vi by moving a test charge q0 along the path icf. • Idea: Integrate Eds along the path connecting ic then cf. (Imagine that we move a test charge q0 from i to c and then from c to f.) 184 Lecture 9

30. Example - Charge moves in E field distance = sqrt(2) d by Pythagoras 184 Lecture 9

31. Clicker Question Quick: DV is independent of path. Explicit: DV = -  E . ds =  E ds = - Ed • We just derived Vf-Vi for the path i -> c -> f. What is Vf-Vi when going directly from i to f ? A: 0 B: -Ed C: +Ed D: -1/2 Ed 184 Lecture 9