ELECTRICITY PHY 1013 S ELECTRIC FIELDS Gregor Leigh

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

PHY 1013 S ELECTRICITY 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. 2

PHY 1013 S ELECTRICITY ELECTRIC FIELDS 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: Electric field: A region in which a particle of mass experiences a gravitational force. A region in which a charged particle experiences an electrostatic force. 3

PHY 1013 S ELECTRICITY ELECTRIC FIELDS THE GRAVITATIONAL FIELD Hence: …this is the gravitational field due to m 1, written , … …acting on m 2 According to Faraday… I. e. : and gravitational field strength At Earth’s surface: [g. Earth = 9. 8 N/kg or m/s 2] 4

PHY 1013 S ELECTRICITY ELECTRIC FIELDS GRAVITATIONAL FIELD STRENGTH Notes: 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. 5

PHY 1013 S ELECTRICITY ELECTRIC FIELDS 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? ) 6

PHY 1013 S ELECTRICITY ELECTRIC FIELDS ELECTRIC FIELD LINES + – 7

PHY 1013 S ELECTRICITY ELECTRIC FIELDS 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 + + – – – 8

PHY 1013 S ELECTRICITY ELECTRIC FIELDS 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 Direction of (electric field strength): : given by the direction of the force experienced by a (positive) test charge. 9

PHY 1013 S ELECTRICITY ELECTRIC FIELDS THE ELECTRIC FIELD VECTOR, Consider two charges q 1 and q 2: q 1 q 2 The field due to q 1 exerts a force on q 2: And the field due to q 2 exerts a force on q 1: (Note: Fon q 1 = Fon q 2 … but E 1 = E 2 only if q 1 = q 2) 10

PHY 1013 S ELECTRICITY ELECTRIC FIELDS 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: 11

PHY 1013 S ELECTRICITY ELECTRIC FIELDS FIELD DUE TO A POINT 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. Alternatively, in terms of the unit vector , which points straight outward from the source charge, : 12

PHY 1013 S ELECTRICITY ELECTRIC FIELDS VECTOR FIELD DIAGRAMS Fields can also be represented graphically by drawing field vectors at a few select points… Notes: 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. 13

PHY 1013 S ELECTRICITY ELECTRIC FIELDS 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: 14

PHY 1013 S ELECTRICITY ELECTRIC FIELDS ELECTRIC FIELD DUE TO MULTIPLE POINT CHARGES Pictorial strategy: 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. 15

PHY 1013 S ELECTRICITY ELECTRIC FIELDS TYPICAL ELECTRIC FIELD STRENGTHS Field location Inside a current-carrying wire Field strength [N/C] 10– 2 Near the Earth’s surface 102– 104 Near objects charged by rubbing 103– 106 Electric breakdown in air, resulting in a spark 3 106 Inside an atom 1011 16

PHY 1013 S ELECTRICITY ELECTRIC FIELDS SHAPE OF THE FIELD DUE TO… An isolated point charge: – + 17

PHY 1013 S ELECTRICITY ELECTRIC FIELDS SHAPE OF THE FIELD DUE TO… Two equal, unlike charges, i. e. a dipole: + – 18

PHY 1013 S ELECTRICITY ELECTRIC FIELDS ELECTRIC DIPOLES An electric dipole is a pair of equal and opposite charges +q and –q separated by a small distance s. Temporary dipole – Permanent dipole – formed when a neutral atom is polarised by an external charge. atoms with differing electronegativities combine to form a polar molecule. –– ++ H H O 19

PHY 1013 S ELECTRICITY ELECTRIC FIELDS 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. s +q –q We thus define the dipole moment, , as the vector: = (qs, from the negative to the positive charge) 20

PHY 1013 S ELECTRICITY ELECTRIC FIELDS FIELD DUE TO AN ELECTRIC DIPOLE Although a dipole is neutral overall, it does create a field. y At points along the dipole axis (at large distances from the dipole, i. e. y >> s), it can be shown that: P y + s Q In the plane around the “waist” of the dipole, (for r >> s): x – 21

PHY 1013 S ELECTRICITY ELECTRIC FIELDS DUE TO CONTINUOUS DISTRIBUTIONS OF CHARGE One-dimensional line of charge: The charge per unit of length is known as the linear charge density, Field “around the waist” of a uniformly charged rod: Q L Field due to an infinite line of charge: 22

PHY 1013 S ELECTRICITY ELECTRIC FIELDS DUE TO CONTINUOUS DISTRIBUTIONS OF CHARGE Ring of charge: Q R P z z The electric field on the axis of a charged ring of radius R: 23

PHY 1013 S ELECTRICITY ELECTRIC FIELDS DUE TO CONTINUOUS DISTRIBUTIONS OF CHARGE Disk of charge: The charge per unit of area is known as the surface charge density, Q A R P z z The electric field on the axis of a charged disk of radius R: 24

PHY 1013 S ELECTRICITY ELECTRIC FIELDS DUE TO CONTINUOUS DISTRIBUTIONS OF CHARGE “Infinite” plane of charge: z 25

PHY 1013 S ELECTRICITY ELECTRIC FIELDS DUE TO CONTINUOUS DISTRIBUTIONS OF CHARGE Sphere of charge: Q R The electric field outside a sphere of charge (r R): 26

PHY 1013 S ELECTRICITY ELECTRIC FIELDS POINT CHARGE IN AN ELECTRIC FIELD Include q’s sign. If it is negative, the force q experiences is in the opposite direction to. Millikan’s experiment: R, , q The stationary oil drop carries three extra electrons R = 2. 76 m = 920 kg/m 3 =? 27

PHY 1013 S ELECTRICITY ELECTRIC FIELDS POINT CHARGE IN AN ELECTRIC FIELD Show that… Ink-jet printing: y Calculate y, if… m, q y m, q x L m = 1. 3 10– 10 kg q = – 1. 5 10– 13 C v 0 = 18 m/s E = 1. 4 106 N/C L = 1. 6 cm 28

PHY 1013 S ELECTRICITY ELECTRIC FIELDS DIPOLE IN AN ELECTRIC FIELD A dipole in a uniform electric field experiences no net force. +q –q s/ 2 It does, however, experience torque about its centre of mass… i. e. 29

PHY 1013 S ELECTRICITY ELECTRIC FIELDS DIPOLE IN AN ELECTRIC FIELD Potential energy of a dipole : and minimum when = 0° U( ) set to zero when = 90° maximum when = 180° i. e. 30

PHY 1013 S ELECTRICITY ELECTRIC FIELDS Water molecules and microwave ovens 31