1 / 31

Electric Potential and Work in Electrostatics

This lecture discusses electric potential and electric potential difference in the context of electrostatics. It covers topics such as potential energy, work, and the concept of electric potential. The lecture also includes examples and applications, such as the energy gain of a proton and the use of Van de Graaff generators.

radabaugh
Download Presentation

Electric Potential and Work in Electrostatics

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related