1/23/07184 Lecture 91 PHY 184 Spring 2007 Lecture 9 Title: The Electric Potential

1/23/07184 Lecture 92AnnouncementsAnnouncements  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!)

1/23/07184 Lecture 93 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.

1/23/07184 Lecture 94 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

1/23/07184 Lecture 95 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

1/23/07184 Lecture 96 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

1/23/07184 Lecture 97 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).

1/23/07184 Lecture 98 Electric Potential Difference  V  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

1/23/07184 Lecture 99 Electric Potential Difference (2)  Taking the electric potential energy to be zero at infinity we have where W e,  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  V = V(x)  0

1/23/07184 Lecture 910 The Volt  The commonly encountered unit joules/coulomb is called the volt, abbreviated V, after the Italian physicist Alessandro Volta ( )  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.

1/23/07184 Lecture 911 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? + - = V(+)-V(  ) The potential difference between the two plates is 450 V. The change in potential energy of the proton is  U, and  V =  U / q (by definition of V), so  U = q  V = e[V(  )  V(+)] =  450 eV

1/23/07184 Lecture 912 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 = keV. initial final Conservation of energy  K =  U = eV Because the proton started at rest, K = 1.6x C x 450 V = 7.2x J

1/23/07184 Lecture 913 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 ( ).  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.

1/23/07184 Lecture 914 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.

1/23/07184 Lecture 915 The Tandem Van de Graaff Accelerator  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). Terminal at +10MV C -1 C +6 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.

1/23/07184 Lecture 916 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 12 C 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 energy accelerating toward the terminal The stripped carbon ion with a +6e charge gains energy accelerating away from the terminal 15 MV Tandem Van de Graaff at Brookhaven

1/23/07184 Lecture 917 Example - Energy of Tandem Accelerator (2)

1/23/07184 Lecture 918 Equipotential Surfaces and Lines  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 three dimensions.  We will often take advantage of symmetries in the electric potential and represent the equipotential surfaces as equipotential lines in a plane. Equipotential surface from eight point charges fixed at the corners of a cube

1/23/07184 Lecture 919  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. if d  E General Considerations

1/23/07184 Lecture 920 Electric field lines and equipotential surfaces

1/23/07184 Lecture 921 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

1/23/07184 Lecture 922 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

1/23/07184 Lecture 923 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 positive electric potential.  The blue lines represent negative electric potential.  Close to each charge, the equipotential lines resemble those from a point charge.

1/23/07184 Lecture 924 ELECTRIC DIPOLE

1/23/07184 Lecture 925 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 lines are always perpendicular to the electric field lines.  There are only positive potentials.  Close to each charge, the equipotential lines resemble those from a point charge.

1/23/07184 Lecture 926 TWO POSITIVE CHARGES

1/23/07184 Lecture 927 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 field F = 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

1/23/07184 Lecture 928 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)

1/23/07184 Lecture 929 Example - Charge moves in E field  Given the uniform electric field E, find the potential difference V f -V i by moving a test charge q 0 along the path icf.  Idea: Integrate E  ds along the path connecting ic then cf. (Imagine that we move a test charge q 0 from i to c and then from c to f.)

1/23/07184 Lecture 930 Example - Charge moves in E field distance = sqrt(2) d by Pythagoras

1/23/07184 Lecture 931 Clicker Question  We just derived V f -V i for the path i -> c -> f. What is V f -V i when going directly from i to f ? A: 0 B: -Ed C: +Ed D: -1/2 Ed Quick:  V is independent of path. Explicit:  V = -  E. ds =  E ds = - Ed