1. ELECTRICITY PHY1013S POTENTIAL Gregor Leigh gregor.leigh@uct.ac.za

2. ELECTRICITY ELECTRIC POTENTIALPHY1013S 2 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.

3. ELECTRICITY ELECTRIC POTENTIALPHY1013S 3 GRAVITATIONAL POTENTIAL DIFFERENCE Objects have different potential energies at different points (heights) in a gravitational field. 2 kg 4 kg 80 g 0.5 m 1 m

4. ELECTRICITY ELECTRIC POTENTIALPHY1013S 4 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  U = 0.08  1  9.8 J 0.5 m 1 m 2 kg 4 kg 80 g  U = 4  0.5  9.8 J

5. ELECTRICITY ELECTRIC POTENTIALPHY1013S 5 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. 0.5 m 1 m  U = 9.8 J per 1 kg 2 kg 4 kg 80 g  U = 4.9 J per 1 kg

6. ELECTRICITY ELECTRIC POTENTIALPHY1013S 6  U = 9.8 J per 1 kg  U = 4.9 J per 1 kg GRAVITATIONAL POTENTIAL DIFFERENCE We might call this difference in gravitational potential energy per unit of mass the gravitational potential difference between the two points: 0.5 m 1 m  G = 9.8 J/kg  G = 4.9 J/kg

7. ELECTRICITY ELECTRIC POTENTIALPHY1013S 7 Units: [J/kg] And hence we might (in the interests of obfuscation) talk about the “greggage” between two points in a gravitational field. GRAVITATIONAL POTENTIAL DIFFERENCE 0.5 m 1 m  G = 9.8 G  G = 4.9 G  G = 9.8 J/kg  G = 4.9 J/kg = [greg, G]

8. ELECTRICITY ELECTRIC POTENTIALPHY1013S 8 ELECTRIC POTENTIAL ENERGY Electric potential energy Electrostatic potential energy U But only changes or differences in potential energy are meaningful. As a field does work on a charged particle the particle loses potential energy: 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…”. Notes:  U = U f – U i = –W elec

9. ELECTRICITY ELECTRIC POTENTIALPHY1013S 9 ELECTRIC POTENTIAL DIFFERENCE The difference in the amount of electric potential energy per unit of charge between one point and another in an electric field is known as… differencepotential 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. …the potential difference between those points.

10. ELECTRICITY ELECTRIC POTENTIALPHY1013S 10 ELECTRIC POTENTIAL, V Using infinity as our reference (zero) point, U i = 0, Electric potential (at a point) Potential (??!)  U = U f – U i = U f – 0 = U f = –W  Hence the potential at a point is given by: But what could possibly be meant by “the potential at one point”? or simply: U = –W  and hence:

11. ELECTRICITY but sometimes it is easier to use the electron volt (eV): 1 eV = 1.6  10 –19 J. ELECTRIC POTENTIALPHY1013S 11 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 = qV  [J = C V]…

12. ELECTRICITY ELECTRIC POTENTIALPHY1013S 12 V 1 = 80 V EQUIPOTENTIAL SURFACES An equipotential surface is a collection of points which all have the same potential. 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. c b a V 2 = 60 V V 3 = 40 V

13. ELECTRICITY ELECTRIC POTENTIALPHY1013S 13 EQUIPOTENTIAL CONTOURS – ++ equipotential contours Equipotential surfaces (which lie perpendicular to the field lines) can also be represented as equipotential contours:

14. ELECTRICITY ELECTRIC POTENTIALPHY1013S 14 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… 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 i f + q +

15. ELECTRICITY ELECTRIC POTENTIALPHY1013S 15 And since (from our definition of V ) If i is at infinity (where V i = 0 ), then the potential at any point relative to infinity is CALCULATING THE POTENTIAL FROM THE ELECTRIC FIELD + q i f

16. ELECTRICITY ELECTRIC POTENTIALPHY1013S 16 POTENTIAL DUE TO A POINT CHARGE Letting a test charge move radially inwards from  to P… and hence… q q0q0 P r' r +

17. ELECTRICITY ELECTRIC POTENTIALPHY1013S 17 POTENTIAL DUE TO MULTIPLE POINT CHARGES Hence: The sign of each q i determines the sign of its V i, but the addition is algebraic, not vector! For a continuous charge distribution: Notes: The principle of superposition applies, i.e..

18. ELECTRICITY ELECTRIC POTENTIALPHY1013S 18 P POTENTIAL DUE TO AN ELECTRIC DIPOLE When r >> s … + –q +q z O  s + – The potential at P is… r+r+ r–r– r

19. ELECTRICITY ELECTRIC POTENTIALPHY1013S 19 POTENTIAL DUE TO AN ELECTRIC DIPOLE When r >> s … + –q +q z r+r+ r–r– r s +  r – – r +  s cos  r – r +  r 2 Note: V = 0 for all points in the plane defined by  = 90° r – – r +  s cos  – P O

20. ELECTRICITY ELECTRIC POTENTIALPHY1013S 20 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 dV. The work done by the field E is dW = –q dV and also From which we get  s V V + dV q ++

21. ELECTRICITY ELECTRIC POTENTIALPHY1013S 21 CALCULATING THE ELECTRIC FIELD FROM THE POTENTIAL … E cos  is simply the component of the electric field in the direction of, Taking this direction successively along the three principle axes, we derive the components of E : And if the electric field is uniform… so we can write  s q + V V + dV

22. ELECTRICITY ELECTRIC POTENTIALPHY1013S 22 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. 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: P + q2q2 + q1q1 to charge q 2 r

23. ELECTRICITY ELECTRIC POTENTIALPHY1013S 23 Since and and … ELECTRICAL POTENTIAL ENERGY OF A SYSTEM OF POINT CHARGES + q2q2 + q1q1 r … 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.

24. ELECTRICITY ELECTRIC POTENTIALPHY1013S 24 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). POTENTIAL OF A CHARGED ISOLATED CONDUCTOR Thus the potential is the same at all points on and inside the conductor.

25. ELECTRICITY ELECTRIC POTENTIALPHY1013S 25 For a charged spherical conductor (solid or hollow) of radius r 0 … POTENTIAL OF A CHARGED ISOLATED CONDUCTOR rr0r0 V(r)V(r) E(r)E(r) rr0r0 … And remembering that the electric field is the derivative of the potential…

26. ELECTRICITY ELECTRIC POTENTIALPHY1013S 26 CHARGE DISTRIBUTION ON A NON-SPHERICAL CONDUCTOR The surface of the conductor is an equipotential surface.

27. ELECTRICITY ELECTRIC POTENTIALPHY1013S 27 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.

28. ELECTRICITY ELECTRIC POTENTIALPHY1013S 28 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.

29. ELECTRICITY ELECTRIC POTENTIALPHY1013S 29 CHARGE DISTRIBUTION ON A NON-SPHERICAL CONDUCTOR Equipotential surfaces are closest together where the electric field is strongest.

30. ELECTRICITY ELECTRIC POTENTIALPHY1013S 30 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.

31. ELECTRICITY ELECTRIC POTENTIALPHY1013S 31 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 … the net field inside the conducting material is zero; the net electric field at the surface is perpendicular to the surface. ISOLATED CONDUCTOR IN AN EXTERNAL ELECTRIC FIELD E = 0