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

2. ELECTRICITY ELECTRIC FIELDSPHY1013S 2 ELECTRIC FIELDS Learning outcomes: At the end of this chapter you should be able to… Use a field model to explain the long-range interaction between charges. Determine the shape and strengths of the various electric fields due to specific configurations of charge. Calculate the forces on (and the motion of) point charges and dipoles in each of these fields. Determine the energy necessary to rotate a dipole in a uniform electric field.

3. ELECTRICITY ELECTRIC FIELDSPHY1013S 3 THE CONCEPT OF A FIELD How do some forces (gravity, electrostatics, magnetism) act at a distance? Faraday proposed that the space itself around certain quantities (e.g. mass, charge) is “filled with influence”. It is this “altered space”, called a field, which becomes the agent acting directly on a second body in the field. Gravitational field:A region in which a particle of mass experiences a gravitational force. Electric field:A region in which a charged particle experiences an electrostatic force.

4. ELECTRICITY ELECTRIC FIELDSPHY1013S 4 I.e.: THE GRAVITATIONAL FIELD Hence: According to Faraday… …this is the gravitational field due to m 1, written,… …acting on m 2 At Earth’s surface: and gravitational field strength [ g Earth = 9.8 N/kg or m/s 2 ]

5. ELECTRICITY ELECTRIC FIELDSPHY1013S 5 GRAVITATIONAL FIELD STRENGTH Field strength is proportional to the mass m 1 which creates the field. Larger masses cause stronger gravitational fields. Field strength is inversely proportional to the square of the distance from m 1, but never becomes exactly zero. Field strength does NOT depend on the mass of any other body which experiences the field. In fact, the field exists whether another mass is present to experience it or not. Notes:

6. ELECTRICITY ELECTRIC FIELDSPHY1013S 6 ELECTRIC FIELD LINES A few imaginary “lines of force” (Faraday) are used to represent the existence of a field. The direction of the field is given by the direction in which a test charge (positive) tends to move. Field lines start on + ve charges and end on – ve charges. The tangent to a field line at any point gives the direction of the electric field vector,. Field lines never touch or cross each other. The density of field lines is an indication of the strength of. Field lines leave or arrive at the surface of a conductor at right angles to the surface. (What is the component of parallel to a conducting surface?)

7. ELECTRICITY ELECTRIC FIELDSPHY1013S 7 ELECTRIC FIELD LINES –+

8. ELECTRICITY ELECTRIC FIELDSPHY1013S 8 THE FIELD MODEL Instead of applying Coulomb’s Law directly, it is often more useful to… 1.Determine the electric field at some point, due to a given configuration of “source” charge(s); 2.Calculate the force exerted by the field on an “intruder” charge at that point in the field. –––––– ++++++ q

9. ELECTRICITY ELECTRIC FIELDSPHY1013S 9 THE ELECTRIC FIELD VECTOR, The electric field vector,, (at some point in a field) is defined as the force per unit positive charge at that point: Units: [N/C  V/m] Magnitude of (electric field strength): Direction of :given by the direction of the force experienced by a (positive) test charge.

10. ELECTRICITY ELECTRIC FIELDSPHY1013S 10 Consider two charges q 1 and q 2 : The field due to q 1 exerts a force on q 2 : THE ELECTRIC FIELD VECTOR, And the field due to q 2 exerts a force on q 1 : (Note: F on q1 = F on q2 … but E 1 = E 2 only if q 1 = q 2 ) q1q1 q2q2

11. ELECTRICITY ELECTRIC FIELDSPHY1013S 11 FIELD DUE TO A POINT CHARGE To find the field strength at a point in the field a distance r from a point charge q, we place a test charge q' at that point. According to Coulomb, the magnitude of the force between the charges is: And therefore the electric field strength ( E = F/q' ) is:

12. ELECTRICITY ELECTRIC FIELDSPHY1013S 12 FIELD DUE TO A POINT CHARGE Alternatively, in terms of the unit vector, which points straight outward from the source charge,: The direction of is in the direction of the force on a positive test particle, i.e. away from the source charge q if E is positive, and towards q if E is negative.

13. ELECTRICITY ELECTRIC FIELDSPHY1013S 13 VECTOR FIELD DIAGRAMS Fields can also be represented graphically by drawing field vectors at a few select points… The field exists everywhere – not just at the few representative points in the diagram. The arrow indicates the strength and direction of the field at the point to which it is attached, i.e. at the tail of the vector arrow. Although we use an arrow to represent it, the electric field vector is a point quantity – it does not “stretch” from one point to another. Notes: –

14. ELECTRICITY ELECTRIC FIELDSPHY1013S 14 ELECTRIC FIELD DUE TO MULTIPLE POINT CHARGES The net field produced at any given point as a result of several point charges can be determined by summing the individual electric field vectors (!) at that point: In practice we work with the 3 simultaneous equations:

15. ELECTRICITY ELECTRIC FIELDSPHY1013S 15 Establish a coordinate system and draw in the charges. Identify the point P at which you want to determine the electric field. At P, draw each electric field vector due to each of the source charges. Resolve each electric field vector into x-, y- and z- components. Wherever possible, use symmetry to simplify your calculations. ELECTRIC FIELD DUE TO MULTIPLE POINT CHARGES Pictorial strategy:

16. ELECTRICITY ELECTRIC FIELDSPHY1013S 16 TYPICAL ELECTRIC FIELD STRENGTHS Field locationField strength [N/C] Inside a current-carrying wire Near the Earth’s surface Near objects charged by rubbing Electric breakdown in air, resulting in a spark Inside an atom 10 –2 10 2 –10 4 10 3 –10 6 3  10 6 10 11

17. ELECTRICITY ELECTRIC FIELDSPHY1013S 17 + – SHAPE OF THE FIELD DUE TO… An isolated point charge:

18. ELECTRICITY ELECTRIC FIELDSPHY1013S 18 –+ SHAPE OF THE FIELD DUE TO… Two equal, unlike charges, i.e. a dipole:

19. ELECTRICITY ELECTRIC FIELDSPHY1013S 19 + + – – ELECTRIC DIPOLES An electric dipole is a pair of equal and opposite charges +q and –q separated by a small distance s. Temporary dipole –formed when a neutral atom is polarised by an external charge. Permanent dipole –atoms with differing electronegativities combine to form a polar molecule. + + + H H O

20. ELECTRICITY ELECTRIC FIELDSPHY1013S 20 DIPOLE MOMENT The properties of a dipole are, essentially: the magnitude of the charge on each pole, q ; the distance between the centres of charge, s. We thus define the dipole moment,, as the vector: = ( qs, from the negative to the positive charge) s +q –q–q

21. ELECTRICITY ELECTRIC FIELDSPHY1013S 21 FIELD DUE TO AN ELECTRIC DIPOLE Although a dipole is neutral overall, it does create a field. y x s P Q At points along the dipole axis (at large distances from the dipole, i.e. y >> s ), it can be shown that: y In the plane around the “waist” of the dipole, (for r >> s ): + –

22. ELECTRICITY ELECTRIC FIELDSPHY1013S 22 L The charge per unit of length is known as the linear charge density, Field “around the waist” of a uniformly charged rod: Q Field due to an infinite line of charge: FIELDS DUE TO CONTINUOUS DISTRIBUTIONS OF CHARGE One-dimensional line of charge:

23. ELECTRICITY ELECTRIC FIELDSPHY1013S 23 Ring of charge: R Q The electric field on the axis of a charged ring of radius R : FIELDS DUE TO CONTINUOUS DISTRIBUTIONS OF CHARGE P z z

24. ELECTRICITY ELECTRIC FIELDSPHY1013S 24 Q The electric field on the axis of a charged disk of radius R : FIELDS DUE TO CONTINUOUS DISTRIBUTIONS OF CHARGE Disk of charge: P z z R The charge per unit of area is known as the surface charge density, A

25. ELECTRICITY ELECTRIC FIELDSPHY1013S 25 FIELDS DUE TO CONTINUOUS DISTRIBUTIONS OF CHARGE “Infinite” plane of charge: z

26. ELECTRICITY ELECTRIC FIELDSPHY1013S 26 FIELDS DUE TO CONTINUOUS DISTRIBUTIONS OF CHARGE Sphere of charge: The electric field outside a sphere of charge ( r  R ): Q R

27. ELECTRICITY ELECTRIC FIELDSPHY1013S 27 Include q ’s sign. If it is negative, the force q experiences is in the opposite direction to. POINT CHARGE IN AN ELECTRIC FIELD Millikan’s experiment: The stationary oil drop carries three extra electrons R = 2.76  m  = 920 kg/m 3 = ? R, , q

28. ELECTRICITY ELECTRIC FIELDSPHY1013S 28 POINT CHARGE IN AN ELECTRIC FIELD Ink-jet printing: Calculate y, if… m = 1.3  10 –10 kg q = –1.5  10 –13 C v 0 = 18 m/s E = 1.4  10 6 N/C L = 1.6 cm Show that… y x L m, q y

29. ELECTRICITY ELECTRIC FIELDSPHY1013S 29 DIPOLE IN AN ELECTRIC FIELD +q –q  s/2s/2 i.e. A dipole in a uniform electric field experiences no net force. It does, however, experience torque about its centre of mass…

30. ELECTRICITY ELECTRIC FIELDSPHY1013S 30 DIPOLE IN AN ELECTRIC FIELD Potential energy of a dipole : i.e. and U()U() minimum when  = 0 ° set to zero when  = 90° maximum when  = 180°  

31. ELECTRICITY ELECTRIC FIELDSPHY1013S 31 Water molecules and microwave ovens