ELECTRICITY PHY 1013 S POTENTIAL Gregor Leigh gregor

ELECTRICITY PHY 1013 S POTENTIAL Gregor Leigh gregor. leigh@uct. ac. za

PHY 1013 S ELECTRICITY ELECTRIC POTENTIAL Learning outcomes: At the end of this chapter you should be able to… Distinguish carefully between electrical potential energy, potential difference and potential (and other terminology). Determine the electric potentials at various points in fields due to specific charge distributions, and illustrate these potentials using several graphical representations. Calculate electric potential from electric field & vice versa. Apply the law of conservation of energy to determine the behaviour of charged particles in electric fields. 2

PHY 1013 S ELECTRICITY ELECTRIC POTENTIAL GRAVITATIONAL POTENTIAL DIFFERENCE Objects have different potential energies at different points (heights) in a gravitational field. 1 m 2 kg 4 kg 80 g 0. 5 m 3

PHY 1013 S ELECTRICITY ELECTRIC POTENTIAL GRAVITATIONAL POTENTIAL DIFFERENCE The actual difference in potential energy between the two points depends on the mass being moved. U = 2 1 9. 8 J 2 kg 1 m U = 0. 08 1 9. 8 J 80 g 4 kg U = 4 0. 5 9. 8 J 0. 5 m 4

PHY 1013 S ELECTRICITY ELECTRIC POTENTIAL GRAVITATIONAL POTENTIAL DIFFERENCE But if instead we consider the difference in the potential energy per unit of mass (i. e. for each kilogram) between the two points, we are considering a property of the field. 2 kg 1 m U = 9. 8 J per 1 kg 80 g 4 kg U = 4. 9 J per 1 kg 0. 5 m 5

PHY 1013 S ELECTRICITY ELECTRIC POTENTIAL GRAVITATIONAL POTENTIAL DIFFERENCE We might call this difference in gravitational potential energy per unit of mass the gravitational potential difference between the two points: 1 m G U = 9. 8 J/kg J per 1 kg U = G 4. 9 = J per 4. 9 J/kg 1 kg 0. 5 m 6

PHY 1013 S ELECTRICITY ELECTRIC POTENTIAL GRAVITATIONAL POTENTIAL DIFFERENCE Units: [J/kg] = [greg, G] And hence we might (in the interests of obfuscation) talk about the “greggage” between two points in a gravitational field. 1 m G = 9. 8 G J/kg G G = 4. 9 J/kg G 0. 5 m 7

PHY 1013 S ELECTRICITY ELECTRIC POTENTIAL ENERGY Electric potential energy Electrostatic potential energy Uelec But only changes or differences in potential energy are meaningful. As a field does work on a charged particle the particle loses potential energy: U = Uf – Ui = –W Notes: The energy is the energy of a system of charges, but you will hear “the energy of a particle…”. The work done is done by the force on the particle due to the other charge(s), but you will hear “the work done by the field…”. 8

PHY 1013 S ELECTRICITY ELECTRIC POTENTIAL DIFFERENCE The difference in the amount of electric potential energy per potential unit of charge between one point and another in an electric field is known as… …the potential difference between those points. Electric potential difference Potential difference Units: [J/C = volt, V] Hence electric potential difference is sometimes (colloquially) referred to as the voltage between two points, or “across” a component in a circuit. 9

PHY 1013 S ELECTRICITY ELECTRIC POTENTIAL, V But what could possibly be meant by “the potential at one point”? Electric potential (at a point) Potential (? ? !) Using infinity as our reference (zero) point, Ui = 0, and hence: or simply: U = Uf – Ui = Uf – 0 = Uf = –W U = –W Hence the potential at a point is given by: 10

PHY 1013 S ELECTRICITY ELECTRIC POTENTIAL, V Notes: Electric potential is a property of the field (or, more specifically, it depends on the source charges and their geometry). Though we use a “probe” charge to measure it, like the field itself, potential exists whether an “intruder” charge is there to experience it or not. Electric potential is a scalar quantity. Like potential difference, potential is measured in volts (joules/coulomb). w = q. V [J = C V]… but sometimes it is easier to use the electron volt (e. V): 1 e. V = 1. 6 10– 19 J. 11

PHY 1013 S ELECTRICITY ELECTRIC POTENTIAL EQUIPOTENTIAL SURFACES An equipotential surface is a collection of points which all have the same potential. b V 1 = 80 V a c V 2 = 60 V V 3 = 40 V No net work is done by or against the electric field when a charge moves between two points on the same equipotential surface (whatever route it follows). The work done moving a charge from one equipotential surface to another is independent of the path taken. Equipotential surfaces lie perpendicular to field lines. 12

PHY 1013 S ELECTRICITY ELECTRIC POTENTIAL EQUIPOTENTIAL CONTOURS Equipotential surfaces (which lie perpendicular to the field lines) can also be represented as equipotential contours: + + – equipotential contours 13

PHY 1013 S ELECTRICITY ELECTRIC POTENTIAL CALCULATING THE POTENTIAL FROM THE ELECTRIC FIELD A particle with charge q moves from initial position i to final position f along an arbitrary path in a non-uniform field… i + q + f At any point the force acting on the particle is and the differential work done by the field during a displacement is Integrating over the whole path for the net work done by the field, we get 14

PHY 1013 S ELECTRICITY ELECTRIC POTENTIAL CALCULATING THE POTENTIAL FROM THE ELECTRIC FIELD i q + f And since (from our definition of V) If i is at infinity (where Vi = 0), then the potential at any point relative to infinity is 15

PHY 1013 S ELECTRICITY ELECTRIC POTENTIAL DUE TO A POINT CHARGE Letting a test charge move radially inwards from to P… q 0 P r' r + q and hence… 16

PHY 1013 S ELECTRICITY ELECTRIC POTENTIAL DUE TO MULTIPLE POINT CHARGES The principle of superposition applies, i. e. . Hence: Notes: The sign of each qi determines the sign of its Vi, but the addition is algebraic, not vector! For a continuous charge distribution: 17

PHY 1013 S ELECTRICITY ELECTRIC POTENTIAL DUE TO AN ELECTRIC DIPOLE P z The potential at P is… r+ r +q ++ s O –q r– – When r >> s… 18

PHY 1013 S ELECTRICITY ELECTRIC POTENTIAL DUE TO AN ELECTRIC DIPOLE P z r+ r +q ++ s O –q When r >> s… r– – r+ s cos r– r–r+ r 2 – r– – r+ s cos Note: V = 0 for all points in the plane defined by = 90° 19

PHY 1013 S ELECTRICITY ELECTRIC POTENTIAL CALCULATING THE ELECTRIC FIELD FROM THE POTENTIAL A positive test charge moves along the path interval between two equipotential surfaces. The potential difference between the surfaces is d. V. The work done by the field E is d. W = –q d. V q + s V V + d. V and also From which we get 20

PHY 1013 S ELECTRICITY ELECTRIC POTENTIAL CALCULATING THE ELECTRIC FIELD FROM THE POTENTIAL … E cos is simply the component of the electric field in the direction of , so we can write q + s V V + d. V Taking this direction successively along the three principle axes, we derive the components of E: And if the electric field is uniform… 21

PHY 1013 S ELECTRICITY ELECTRIC POTENTIAL ELECTRICAL POTENTIAL ENERGY OF A SYSTEM OF POINT CHARGES The electric potential energy of a system of fixed charges is equal to the work done by an external agent in assembling the system. q 1 P + to charge q 2 r While q 2 is still at , the potential at the position P which will be occupied by q 2 is Bringing q 2 in from to P, requires work: 22

PHY 1013 S ELECTRICITY ELECTRIC POTENTIAL ELECTRICAL POTENTIAL ENERGY OF A SYSTEM OF POINT CHARGES q 1 q 2 + + r Since and … … the potential energy of the pair of charges is thus If q 1 and q 2 are unlike charges the work is done by the field, and the system has negative potential energy. 23

PHY 1013 S ELECTRICITY ELECTRIC POTENTIAL OF A CHARGED ISOLATED CONDUCTOR Excess charge on an isolated conductor distributes itself on the surface of the conductor in such a way that the field inside the conducting material is zero (regardless of whether the conductor has an internal cavity – which may or may not contain a net charge). Thus the potential is the same at all points on and inside the conductor. 24

PHY 1013 S ELECTRICITY ELECTRIC POTENTIAL OF A CHARGED ISOLATED CONDUCTOR For a charged spherical conductor (solid or hollow) of radius r 0 … … And remembering that the electric field is the derivative V(r) of the potential… E(r) r 0 r 25

PHY 1013 S ELECTRICITY ELECTRIC POTENTIAL CHARGE DISTRIBUTION ON A NON-SPHERICAL CONDUCTOR The surface of the conductor is an equipotential surface. 26

PHY 1013 S ELECTRICITY ELECTRIC POTENTIAL CHARGE DISTRIBUTION ON A NON-SPHERICAL CONDUCTOR The further one moves away from a tiny conductor, the more the equipotential surfaces resemble those around a point charge, i. e. they become spherical. 27

PHY 1013 S ELECTRICITY ELECTRIC POTENTIAL CHARGE DISTRIBUTION ON A NON-SPHERICAL CONDUCTOR In order to “morph” into spheres, equipotential surfaces near small-radius convex surface elements have to be closer than they are near “flatter” parts of the surface. 28

PHY 1013 S ELECTRICITY ELECTRIC POTENTIAL CHARGE DISTRIBUTION ON A NON-SPHERICAL CONDUCTOR Equipotential surfaces are closest together where the electric field is strongest. 29

PHY 1013 S ELECTRICITY ELECTRIC POTENTIAL CHARGE DISTRIBUTION ON A NON-SPHERICAL CONDUCTOR Thus on an isolated conductor the concentration of charges and hence the strength of the electric field is greater near sharp points where the curvature is large. 30

PHY 1013 S ELECTRICITY ELECTRIC POTENTIAL ISOLATED CONDUCTOR IN AN EXTERNAL ELECTRIC FIELD For an isolated, uncharged conductor in an external field, the free charges (electrons) distribute themselves on the surface of the conductor in such a way that … E=0 the net field inside the conducting material is zero; the net electric field at the surface is perpendicular to the surface. 31