INTRODUCTION ELECTROSTATIC POTENTIAL ENERGY. Introduction. Electrostatic potential energy. Electric potential. for a system of point charges

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1 Chapter 4 ELECTRIC POTENTIAL Introduction Electrostatic potential energy Electric potential for a system of point charges for a continuous charge distribution Why determine electic potential? Determination of E field from electric potential Circulation of a static electric field Summary INTRODUCTION The concept of electric field has been defined as: E = F The last two lectures focussed on the calculation of the electric field produced by charge distributions basedontwomethods. 1) Coulomb s Law: Discrete charges: E = X = cr 0 Continuous charges: In cartesian coordinates the E field at the point () is written as an integral over the charge distribution at 0 ( ): E () = 1 ( ) dr ) Gauss s Law The surface integral of the flux of the vector E field over a Gaussian surface is related to the enclosed charge by: I Φ = E S = 1 0 Figure 1 Work done against a force field moving from point a to b. Gauss s law was shown to be a simple method for calculating electric fields for systems having symmetry. Gauss Law is a more general law of electromagnetism than Coulomb s law. Gauss s Law applies to both static and rapidly changing electric fields whereas Coulomb s Law is for electrostatics only. Gauss s Law is the first of Maxwell s laws of electromagnetism. This chapter will introduce Maxwell s second law of electrostatics which is a statement about the circulation of the electric field. In addition it provides a third way to determine the electric field E. ELECTROSTATIC POTENTIAL ENERGY In gravitation we learned that the concept of an energy associated with position, or potential energy, is a very useful concept. The potential energy difference between two points r and r, is the work done moving from to against a force field F.Thatis: = (r ) (r )= F l The negative sign in front of the line integral occurs because it takes a positive amount of work to move an object against a force. For example, the stored potential gravitational energy increases by if you lift a mass upwards a distance against the downward gravitational force. In general, this line integral depends on the path taken as well as r and r. For a conservative force, this line integral is independent of the path taken, it only depends on the starting and ending positions, r and r. That is, the potential energy is a local function 27

2 This line integral just depends of the starting and ending positions, for a repulsive force it is negative going from a smaller to larger distance. Path B can be factored into a sequence of infinitessimal radial and tangential steps as shown schematically. Now the line integrals along each of the tangential paths are zero because the path and force are perpendicular. Thus we are left with only the sum of a series of radial integrals. Figure 2 Potential energy difference between r and r along the two paths shown. dependent only on position. The usefulness of gravitational potential is that, since the gravitational force is a conservative force, it is possible to calculate many problems in mechanics using the fact that the sum of the kinetic energy and potential energy is a constant. Coulomb s law will be used to show that the electric force field also is conservative. Proof that the electric force is conservative. Consider the electric field produced by a single point charge q 1 The work done moving a charge q 0 from r to r can be calculated along two paths A and B shown in figure 2. Path A follows a line of constant radius followed by a radial segment, whereas B is an arbitrary path. From Coulomb s law we have that the force of 0 due to point charge 1 is; F 10 = cr 10 Along path A we have that the change in potential energy is given by the line integral: = F l F l Note that along the tangent to a circle, from a to the integral is zero since the path is perpendicular to F and thus the dot product F l =0 The radial integral is straight forward since the radial path is along F and thus the integral is = br br = = X br br = because in the addition all the terms cancel except the first and last terms. Thus we have proven that the electric force is conservative for two point charges. Now use the Principle of Superposition for an electric field produced by a set of charges. F = X =1 F Thelineintegralthencanbewrittenas: = F l = = X =1 X =1 F l Thus we see that the net potential energy difference is the sum of the contributions from each charge producing the electric field. Since each component is conservative, then the total potential energy difference also must be conservative. Note that the electric force is conservative because the force is radial. Similarly, the proof that the gravitational force is conservative is identical except that the product is replaced by 1 0 ELECTRIC POTENTIAL Using F = 0 E gives that the change in potential energy due to moving a charge 0 from to b is: This has units of Joules. = 0 E l 28

3 Note that since the probe charge 0 factors out from the integral it is possible to define a new quantity called electric potential where = = E l 0 That is; electric potential difference is the work that must be done to move a unit charge from a to b with no change in kinetic energy. Be careful not to confuse electric potential energy difference and electric potential difference,thatis, has units of energy, Joules, while has units of Joules/Coulomb or volts. N.B. I have chosen to use the symbol to designate electric potential, and for volume to avoid confusion. The electric potential is a property of the electric field; it is given as minus the line integral of the electric field from a to b. The change in potential energy for movingacharge 0 from to isgivenintermsof electric potential by: = 0 Note that the electric field is conservative, since the potential energy difference is independent of the path taken. Therefore the electric potential difference also must be independent of the path taken. Superposition and potential Previously it was shown that the electric force is conservative for the superposition of many charges. To recap, if E = E 1 + E 2 + E 3,then = E l = = Σ E1 l E2 l E3 l Thus, like electric potential energy, electric potential is a simple additive scalar field because the Principle of Superposition applies. Units: The unit of electric potential is the volt = Also from the line integral definition of electric potential and, it can be seen that electric field has units of volts/meter as well as Newtons/Coulomb. The volt is a commonly used unit in everyday life. The greater the voltage, the greater the potential energy difference in moving a charge through this voltage. The gravitational analog, gravitational potential, between two points differing by in height, is. Clearly, the greater or, the greater the energy released by the Figure 3 Electric field and equipotentials for a charge distribution. gravitational field when dropping a body through the height. Reference potentials: Note that only differences in potential energy,, and electric potential,, are meaningful, the absolute values depend on some arbitrarily chosen reference. However, often it is useful to measure electric potential with respect to a certain fixed reference point just as gravitational potential is usually referenced relative to sea level. For electricity, one often references electric potential relative to the potential of the earth. Thus if point A is at +10 volts relative to earth, and point B is at +50 volts relative to earth, then the electric potential difference between A and B is that B is +40 volts higher than A. For point charges it is more convenient to choose the reference at = where the electric field falls to zero. Equipotentials Note that the scalar product E l is zero if the vectors E and l are mutually perpendicular. Thus the electric potential is constant along surfaces perpendicular to E. Lines of constant electric potential are called equipotentials. Figure 3 illustrates equipotentials and the electric field distribution for a given charge distribution. VFORASYSTEMOFPOINT CHARGES Using Coulomb s law it was shown that the change in electric potential energy from moving 0 in the field of a single point charge 1 from point a to b is: =

4 Figure 4 The equipotentials (dashed) and field lines (solid) for an electric dipole. Thusthechangeinelectricpotentialduetotheelectric field from a point charge 1 is: = where 1 and 1 are the distances of a and b from 1. As shown before, the electric potential due to a system of point charges is additive because of superposition, that is, = X =1 = X = If the reference, is chosen at infinity, then one obtains that the electric potential at the point b, relative to infinity, for a ensemble of fixed discrete charges, is = X =1 Example: Electric dipole Performing the above summation for the equal but opposite sign charges for an electric dipole results in the equipotentials shown in figure 4 as will be discussed next week. Note that the electric field is perpendicular to the equipotentials. V DUE TO CONTINUOUS CHARGE DISTRIBUTIONS Suppose charge is distributed over a volume with a density at any point within the volume. The electric potential at any field point due to an element of 30 Figure 5 Calculation of the electric potential at a distance r from the center of a spherical shell of surface charge density and radius R. The equipotential and electric field are shown. charge = at a point p is given by: ( 0 ) 0 = 4 0 Again, the integral is over a scalar quantity. Sinceelectricpotential is a scalar quantity, it is easier to compute than is the vector electric field E. Example: Uniformly-charge spherical shell, radius R, charge density Consider a circular element, between and + which will have a charge of =2 sin At distance from the center of the shell, and outside of the shell, the potential from this circular element of charge, relative to = is = 2 sin = 4 4 where 2 = cos The total potential is obtained by integrating over the whole sphere; = Since 2 =2 sin then = sin 4 = 2 = 4

5 For inside the spherical charged shell the potential is: = 2 + = 2 = 4 Note that outside of the spherical charge shell, the potential has a 1 radial dependence, whereas it is a constant inside of the spherical shell since the electric field inside is zero. WHY DETERMINE ELECTRIC POTENTIAL? You may question the usefulness of electric potential since the electric forces are given directly in terms of the electric field There are two main applications of electric potential, for calculation of electrostatic potential energy and for determining the electric field. 1) Electrostatic potential energy Energy is stored in an electric field, as is dramatically demonstrated by the energy released in a lightning strike, the energy transmitted through space by electromagnetic waves, and the energy stored in atomic and molecular binding. Since the electric force is conservative, one can solve many problems using energy conservation similar to the situation for the gravitational field. The electric potential energy difference is given in terms of the charge 0 and potential difference by: 2) Electric field Electric potential obviously is of use if you wish to compute the difference in electric potential energy between two locations as discussed above. However, there is a more important reason for computing electric potential in that it is directly related to the electric field. You are familiar with the use of gravitational potential energy curves on contour maps to indicate gravitational forces. For example, figure 6 shows a contour map for for the Minarets - Deadhorse Lake region. The closeness of the equipotentials indicates the steepness of the slope which points perpendicular to the equipotentials. Similarly, the electric field is given by the closeness of the electric equipotentials as described next. DETERMINATION OF E FROM The electric field and electric potential are directly related by: = E l For an arbitrary infinitessimal element distance l the change in electric potential is = E l Using cartesian coordinates both E and l can be written as E = b i + b j + b k = 0 Electron-volt unit of energy The MKS unit of electric energy is given in Joules, that is, the same unit as used for mechanics. However, this is an inconveniently large unit for molecular, atomic and nuclear physics where one is dealing with the charge on the proton or electron of = ± = ± Therefore, for such cases, it is convenient to define an alternate unit of energy called the electron-volt (ev) which is the energy to move a charge through a potential difference of 1 volt: In the hydrogen atom, the electron is bound to the proton by 136. Chemistry normally deals with energies about 1, atomic physics ranges from 1 to 10 nuclear physics deals in the range of 1 to 1000 (10 9 ), while the scale of high-energy physics extends upwards from 1 (10 9 ) l = b i + b j + b k Taking the scalar product gives: = E l = Using differential calculus we know that one can express the change in potential in terms of partial derivatives by: = + + By association, this gives: = = = Thus on each axis, the electric field is minus the gradient of the electric potential. In three dimensions, the electric field is minus the total gradient of the electric potential. That is, the electric field is just the 31

6 Figure 7 Circulation of the electric field. Figure 6 Gravitational equipotentials for the Minarets- DeadHorseLakeregionoftheSierramountains,CA. gradient of the electric potential which is it is perpendicular to the equipotentials. Skiers are familiar with the concept of gravitational equipotentials and the fact that the line of steepest descent, and thus maximum acceleration, is perpendicular to gravitational equipotentials of constant height. The gradient of the scalar function is written in mathematics as which is a vector quantity. Namely: E = In cartesian coordinates this can be written as: E = b i +b j + k b Knowing the electric potential allows calculation of the electric field. The gradient of, also can be expressed in spherical or cylindrical coordinates. CIRCULATION OF A STATIC ELECTRIC FIELD Gauss s law, which is one of Maxwell s equations, was introduced last lecture. It related the flux of the electric field out of a closed surface to the net charge enclosed by this surface. The second of Maxwell s equations for electrostatics addresses the circulation of the electric field. At the beginning of this chapter it was shown that the electric field is conservative, that is is independant of the path taken between a and b. Therefore, the electric potential: = = E l 0 is independent of the path taken between two points a and b. Consider two possible paths between a and basshowninfigure 10. The line integral from a to b via route 1 is equal and opposite to the line integral back from b to a via route 2 if the electric field is conservative as shown earlier A better way of expressing this is that the line integral of is zero around any closed path. That is, the line integral between a and b, via path 1, and returning back to a, via 2, are equal and opposite. That is; I E l = E l + E l = E l E l =0 Therefore the net line integral for a closed loop is zero. I E l =0 This is a measure of the circulation of the electric field. The fact that it is zero is a statement that the electric field is radial for a point charge. In summary we can rewrite the laws of electrostatics, as given by Coulomb s law, in terms of the net flux over a closed surface for the vector E field: I Φ = E S = 1 0 and the net circulation for a closed loop: I E l =0 The zero circulation is a statement that the electric field in Coulomb s law is radial. The flux relation tells you that the field for a point charge has 1 dependence. The concepts of flux and circulation provide a 2 more general statement of Coulomb s law that is applicable to both static and non static electric fields. Later you will see that for changing electromagnetic fields, the circulation equation is modified with crucially important consequences. 32

7 Electric potential SUMMARY The concept of electric potential has been introduced where electric potential difference is given as: = = E l 0 The derivation of E from was shown. In particular, E is given by the gradient of V, that is: E = The application of electric potential to systems of electric conductors was discussed. Maxwells equations for electrostatics The laws of electrostatics have been rewritten in terms of the concepts of flux and circulation of a vector field. Gauss s law gives the net flux for a closed surface: I Φ = E S = 1 0 The flux relation tells you that the field for a point charge has 1 dependence. 2 The net circulation for a closed loop: I E l =0 The zero circulation is a statement that the electric field in Coulomb s law is radial. The concepts of flux and circulation provide a more general statement of Coulomb s law that is applicable to both static and non static electric fields. You will see later that, for changing electromagetic fields, the circulation equation is modified with crucially important consequences. Three different approaches have been presented for calculating the electric field. In order of simplicity, there is; Gauss s law which gives an elegant quick solution for systems having symmetry, electric potential which is more generally useful, and finally, the brute force direct evaluation of the electric field using Coulomb s law. Reading assignment: Giancoli, Chapter 23 33