Electricity and Magnetism

Transcription

1 Electricity and Magnetism Electricity and magnetism have been known for thousands of years The ancient Greeks knew that a piece of amber rubbed with fur would attract small, light objects The word for electron and electricity derived from the Greek word for amber, ηλεκτρον Naturally occurring magnetic materials called lodestones were used as early as 300 BC to construct compasses The relationship between electricity and magnetism was not known until the middle of the 19 th century Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 10-ene-13 Tema VIII. Electrostática (1) 1

2 Fundamental Forces of Nature The force of gravity has been known since the time of Newton Late 17th century In the 0th century, two more forces were discovered The weak force and the strong force Around 1970 the electromagnetic force and the weak force were unified The electroweak force 1979 Nobel prize in physics for Weinberg, Salam, and Glashow Currently physicists are working to unify the electroweak force and the strong force Gravity remains a puzzle although it was identified first 10-ene-13 Tema VIII. Electrostática (1)

3 The Four Forces In our model of the world, the four fundamental forces work by exchanging elementary particles Gravity - graviton (has not been observed yet) Electromagnetic - photon Weak - W and Z bosons (observed in 1983) Strong gluons (observed in 1979) Thus forces can act a distance without touching The Sun can attract the Earth from 93 million miles away Magnet can attract metal 10-ene-13 Tema VIII. Electrostática (1) 3

4 The Fundamental Forces Range: Infinite Infinite m 0.1% of the diameter of a proton m diameter of medium-sized nucleus 10-ene-13 Tema VIII. Electrostática (1) 4

5 Gravitational and Electric Forces For gravity we defined a gravitational force F(r) = G m 1m r And a gravitational potential U(r) = G m 1 m r We will do the same for the electric force and the electric potential We will introduce the concept of an electric field to help us understand the electromagnetic force 10-ene-13 Tema VIII. Electrostática (1) 5

6 Electric Charge Everyday example: When walking on a carpet on a dry winter s day and then touching a door knob, one often experiences a spark This process is called charging Charging: negatively charged electrons move from the atoms and molecules of the carpet to the soles of our shoes, to the body Spark: The built-up charge discharges through the metal of the door knob Similar phenomenon involving wind, rain and ice produces lightning 10-ene-13 Tema VIII. Electrostática (1) 6

7 Charge () Normally objects around us do not seem to carry a net charge They have equal amounts of positive and negative charge and are thus electrically neutral Negative charge: an excess of electrons Positive charge: a deficit of electrons Demo: If we rub a plastic rod with fur, the rod will become charged If we bring two charged plastic rods together, they will repel each other If we rub a glass rod with silk, the rod will become charged If we bring together a charged plastic rod and a charged glass rod, they will attract each other 10-ene-13 Tema VIII. Electrostática (1) 7

8 Measuring Charge: The Electroscope Shield Metal Electric repulsion Metal deflection arm Metal rod Gravity The glass and the plastic rod have opposite charge 10-ene-13 Tema VIII. Electrostática (1) 8

9 Explanation: Explanation of the Demos Electrons are transferred from the fur onto the plastic rod. This rod now carries a negative charge. Electrons are transferred from the glass rod onto the silk. The glass rod now carries a positive charge (electrons are missing). The electroscope shows the presence of charge. 10-ene-13 Tema VIII. Electrostática (1) 9

10 Law of Charges This result leads to the Law of Charges Like charges repel and opposite charges attract Note that electricity is different from gravitation, in which the force is always attractive m1 m 10-ene-13 Tema VIII. Electrostática (1) 10

11 Static Cling What is the force between an electrically charged object (q) and a neutral object (0)? Observe: It is always attractive Why? Polarization q 10-ene-13 Tema VIII. Electrostática (1) 11

12 The Unit of Charge The unit of charge is the coulomb, abbreviated C [named after Charles-Augustin de Coulomb ( )]. The coulomb is defined in terms of the SI unit for electric current, the ampere, abbreviated A [named after Andre-Marie Ampere ( )]. The ampere is a basic SI unit like the meter, the second, and the kilogram. The unit of charge is defined as 1 C = 1 A s 10-ene-13 Tema VIII. Electrostática (1) 1

13 Charge of an Electron We can define the unit of charge in terms of the charge of one electron An electron is an elementary particle with charge q = -e where e = C A proton is a particle with q = +e e = C 10-ene-13 Tema VIII. Electrostática (1) 13

14 Coulomb of Charge A full coulomb is a very large amount of charge! A lightning discharge can contain 10 s of coulombs The number of electrons required to produce 1 coulomb of charge is N e = 1 C C = Because a coulomb is a large amount of charge, everyday examples of static electricity typically involve 1 microcoulomb = 1 µc = 10-6 C 1 nanocoulomb = 1 nc = 10-9 C 1 picocoulomb = 1 pc = 10-1 C 10-ene-13 Tema VIII. Electrostática (1) 14

15 Charge Conservation Benjamin Franklin ( ) introduced the idea of positive and negative charge (amber or plastic is negative). Franklin also proposed that electric charge is conserved. For example,when a plastic rod is charged by rubbing it with a fur, charge is neither created nor destroyed, but instead electrons are transferred to the rod leaving a net positive charge on the fur. Law of charge conservation The total charge of an isolated system is strictly conserved. This law adds to our list of conservation laws: conservation of energy, conservation of momentum, and conservation of angular momentum The total charge is constant. 10-ene-13 Tema VIII. Electrostática (1) 15

16 Example - Net Charge Supongamos que queremos crear una carga positiva de 10 µ C en un bloque de cobre de,00 kg de masa. Qué fracción en % de electrones deberíamos quitar en el bloque de cobre? The atomic weight of copper is grams per mole The number of copper atoms is (.00 kg) atoms/mole N = atom kg/mole ( ) = = The atomic number of copper is 9, which means there are 9 electrons per atom N e, atom = 9 N = = atom 5 6 The number of electrons in 10 µc is Percentage removed is N e,10µc = C C = Ne,10 µ C % = 100 = 100 = % 6 N e, atom 10-ene-13 Tema VIII. Electrostática (1) 16

17 Insulators and Conductors Superconductors Semiconductors 10-ene-13 Tema VIII. Electrostática (1) 17

18 Summary There are two kinds of electric charge positive and negative. Law of Charges Like charges repel and opposite charges attract The unit of charge is the coulomb defined as 1 C = 1 A s Law of charge conservation The total charge of an isolated system is strictly conserved. 10-ene-13 Tema VIII. Electrostática (1) 18

19 Electrostatic Charging There are two ways to charge an object Conduction Induction 10-ene-13 Tema VIII. Electrostática (1) 19

20 Charging by Conduction Electroscope We brought charge onto the electroscope through contact with the electrode. 10-ene-13 Tema VIII. Electrostática (1) 0

21 Induction The presence of the positively charged rod leads to a redistribution of charge (a kind of polarization). It pulls electrons up to the electrode. 10-ene-13 Tema VIII. Electrostática (1) 1

22 Charging by Induction The presence of the positively charged rod leads to a redistribution of charge Grounding pushes positive charge to Earth (or rather pulls electrons from Earth!) leaving the electroscope negative. ground 10-ene-13 Tema VIII. Electrostática (1)

23 Electric Force - Coulomb s Law Consider two electric charges: q 1 and q The electric force F between these two charges separated by a distance r is given by Coulomb s Law F = kq q 1 r The constant k is called Coulomb s constant and is given by k = Nm /C 10-ene-13 Tema VIII. Electrostática (1) 3

24 Coulomb s Law () The coulomb constant is also written as k = 1 4π ε 0 where ε 0 = C Nm ε 0 is the electric permittivity of vacuum A fundamental constant of nature F = 1 q 1 q 4πε 0 r 10-ene-13 Tema VIII. Electrostática (1) 4

25 Example: Force between Two Charges What is the force between two 1 C charges 1 meter apart? F = 1 q 1 q 4πε 0 r 9 N m F = C 1 C 1 C 9 = N ( 1 m) which is the weight of 450 Space Shuttles at launch 10-ene-13 Tema VIII. Electrostática (1) 5

26 Electric Force The electric force is given by The electric force, unlike the gravitational force, can be positive or negative If the charges have opposite signs, the force is negative Attractive If the charges have the same sign, the force is positive Repulsive F = k q 1q r ene-13 Tema VIII. Electrostática (1) 6

27 Electric Force Vector Electric force in vector form y q 1 r 1 r r r r r = r r 1 r r r q r r r rˆ = = 1 r r r x Fuerza sobre q Fuerza sobre q 1 r F = k q 1q r ˆr r F 1 = k q 1q r ( ˆr ) 10-ene-13 Tema VIII. Electrostática (1) 7

28 Superposition Principle La fuerza neta que actúa sobre cualquier carga es la suma vectorial de las fuerzas debidas a las cargas restantes de la distribución r r r r F = F + F + L+ F 1, net 1, 1,3 1, n F = F + F F F = F + F F F = F + F F 1x 1,, x 1,3, x 1, n, x 1y 1,, y 1,3, y 1, n, y 1z 1,, z 1,3, z 1, n, z 10-ene-13 Tema VIII. Electrostática (1) 8

29 Example - The Helium Nucleus El núcleo de un átomo de helio tiene dos protones y dos neutrones. Cuál es la magnitud de la fuerza eléctrica entre los dos protones en el núcleo de helio? The distance between the two protons is approximately m Each proton has charge q = C The force is given by F = k q q 1 N m = r C ( C) = 58 N ( m) 10-ene-13 Tema VIII. Electrostática (1) 9

30 Example - The Helium Nucleus () Qué pasa si la distancia r entre los protones se duplica? F r = k q q 1 ( r) = k q q 1 F = 1 r F 4 r ( ) 4r with F = k q q 1 r r Ley del inverso del cuadrado: Si la distancia se duplica entonces la fuerza se reduce por un factor 4. F r = 1 ( 4 58 N )= 14.5 N 10-ene-13 Tema VIII. Electrostática (1) 30

31 Clicker Quiz: Equilibrium Position Considera dos cargas colocadas en el eje x A x 1 B x C q 1 = 0.15 µc x 1 = 0.0 m 1 µ 1 q = 0.35 µc x = 0.40 m Dónde tenemos que poner una tercera carga para que esa carga pueda estar en equilibrio? A: to the left of charge 1 B: in the middle between the two charges C: to the right of charge 10-ene-13 Tema VIII. Electrostática (1) 31

32 Clicker Quiz: Equilibrium Position () x 1 x A: x 3 <x 1 Here the forces from q 1 and q will always point in the same direction (to the left for a positive test charge) No equilibrium C: x <x 3 Here the forces from q 1 and q will always point in the same direction (to the right for a positive test charge) No equilibrium 10-ene-13 Tema VIII. Electrostática (1) 3

33 Clicker Quiz: Equilibrium Position (3) x 1 x q 3 B: x 1 < x 3 < x Here the forces from q 1 and q can balance q 1 = 0.15 µc x 1 = 0.0 m q q k = k q = 0.35 µc x = 0.40 m q q ( x x ) ( x x ) q 1 (x 3 x 1 ) = q (x x 3 ) q 1 (x x 3 ) = q (x 3 x 1 ) q 1 (x x 3 ) = q (x 3 x 1 ) x 3 = q 1 x + q x 1 q 1 + q x 3 = q 1x + q x 1 q 1 + q = 0.15 µc (0.4 m) 0.15 µc µc = 0.16 m 10-ene-13 Tema VIII. Electrostática (1) 33

34 Example - Charged Balls (1) Consideremos dos bolas idénticas cargadas que cuelgan del techo por hilos de igual longitud l = 1,5 m (en equilibrio). Cada bola tiene una carga de 5 µ C. Las bolas cuelgan separadas un ángulo θ, θ = 5 con respecto a la vertical. Cuál es la masa de las bolas? Think Three forces act on each ball: Coulomb force F c, gravity F g, and the tension of the string T The coulomb force is horizontal and must be repulsive to keep the balls apart The gravitational force points down The tension T is in the direction of the string The balls are in equilibrium, which means the three forces are balanced 10-ene-13 Tema VIII. Electrostática (1) 34

35 Example - Charged Balls () Sketch Add the three forces to the drawing Define distance d between the balls Make a separate free body diagram for one of the charged balls Define x-y coordinate system 10-ene-13 Tema VIII. Electrostática (1) 35

36 Example - Charged Balls (3) Research The condition of static equilibrium tells us that the sum of the x-components of the three forces acting on the ball must equal zero and the sum of y-components of these forces must equal zero The sum of the x-components of the forces is T sinθ F c = 0 T is magnitude of the string tension θ is the angle of the string relative to the vertical F C is the magnitude of the Coulomb force The sum of the y-components of the forces is T cosθ F g = 0 The force of gravity, F g, is just the weight of the charged ball F g = mg m is the mass of the charged ball 10-ene-13 Tema VIII. Electrostática (1) 36

37 Example - Charged Balls (4) The electric force between the two balls is F c = k q d d is the distance between the two balls We can express d in terms of l and θ sin θ = d / l We can then rewrite the electric force in terms of θ and l q F = k = k c ( lsinθ ) q 4l sin θ 10-ene-13 Tema VIII. Electrostática (1) 37

38 Simplify Example - Charged Balls (5) T sinθ = F c T cosθ = F g Thus we eliminate the string tension and get tanθ = F c F g Putting in the force of gravity and the electric force, q k tanθ = 4l sin θ = mg Solving for m, we get m = kq 4mgl sin θ kq gl θ θ 4 sin tan 10-ene-13 Tema VIII. Electrostática (1) 38

39 Example - Charged Balls (6) Calculate Putting in our numerical values we obtain m = ( N m /C ) 5.0 µc ( ) 4 ( 9.81 m/s )1.50 ( m) sin ( 5.0 )tan( 5.0 ) = kg Round We report our result to three significant figures Double-check m = kg Para comprobar nuestra respuesta, hacemos la aproximación senθ tanθ θ y cosθ 1, de modo que T mg y las componentes x de las fuerzas son q T sinθ mgθ = Fc = k k d q ( lθ ) 10-ene-13 Tema VIII. Electrostática (1) 39

40 Example - Charged Balls (7) The mass of the ball in our double-check is then m ( N m /C )( 5.0 µ C) ( )( ) ( ) kq = = = 3 3 4gl θ m/s 1.50 m rad kg Which is close to our answer m = kg 10-ene-13 Tema VIII. Electrostática (1) 40

41 Example - Forces between Electrons What is relative strength of the electric force compared with the force of gravity for two electrons? F electric = k q e r F gravity = G m e r F electric F gravity = kq e Gm = ( N m / C )( C) = 4. e ( N m /kg )( kg) Gravity is irrelevant for atomic and subatomic processes the electric force is much much stronger. But sometimes gravity is most important; e.g., the motion of the planets. Why? 10-ene-13 Tema VIII. Electrostática (1) 41

42 Example - Four Charges Considere cuatro cargas colocadas en las esquinas de un cuadrado con lados de longitud 1,5 m, como se muestra a la derecha. Cuál es la magnitud de la fuerza eléctrica sobre q 4 resultante de la fuerza eléctrica de los restantes tres cargas? Set up x-y coordinate system with its origin located at q y x-direction F x = k q 1q 4 + k q q 4 d d ( ) cos 45 = kq 4 q d 1 + q cos 45 x y-direction F y = k q q 4 ( d) sin 45 + k q 3q 4 = kq 4 q d d sin 45 + q 3 10-ene-13 Tema VIII. Electrostática (1) 4

43 Example - Four Charges () F = F x + F y F = kq 4 q d 1 + q cos 45 + kq 4 q d sin 45 + q 3 F = kq 4 q d 1 + q cos 45 + q sin 45 + q 3 y q sin 45 = q cos 45 =.50 µc = µc x F = ( ) 4.50 µc F = N ( ) ( 1.50 µc µc) + ( µc 3.50 µc) ( 1.5 m) 10-ene-13 Tema VIII. Electrostática (1) 43