Electric Charges, Forces, and Fields

1. Electric Charges,Forces, andFields

2. Electric Charges • Electric charge is a basic property of matter • Two basic charges • Positive and Negative • Each having an absolute value of • 1.6 x 10-19 Coulombs • Experiments have shown that • Like signed charges repel each other • Unlike signed charges attract each other • For an isolated system, the net charge of the system remains constant • Charge Conservation

3. Two basics type of materials • Conductors • Materials, such as metals, that allow the free movement of charges • Insulators • Materials, such as rubber and glass, that don’t allow the free movement of charges

4. Coulomb’s Law • Coulomb found that the electric force between two charged objects is • Proportional to the product of the charges on the objects, and • Inversely proportional to the separation of the objects squared k being a proportionality constant, having a value of 8.988 x 109 Nm2/c2

5. is a unit vector pointing from object 1 to object 2 q1 q2 Electric Force As with all forces, the electric force is a Vector So we rewrite Coulomb’s Law as This gives the force on charged object 2 due to charged object 1 The direction of the force is either parallel or antiparallel to this unit vector depending upon the relative signs of the charges

6. Electric Force The force acting on each charged object has the same magnitude - but acting in opposite directions (Newton’s Third Law)

7. Q2 Q2 d23 d12 g Q3 Q1 Example 1 • A charged ball Q1 is fixed to a horizontal surface as shown. When another massive charged ball Q2 is brought near, it achieves an equilibrium position at a distance d12directly above Q1. • When Q1 is replaced by a different charged ball Q3, Q2 achieves an equilibrium position at a distance d23 (< d12) directly above Q3. For 1a and 1b which is the correct answer 1a: A)The charge of Q3 has the same sign of the charge of Q1 B) The charge of Q3 has the opposite sign as the charge of Q1 C) Cannot determine the relative signs of the charges of Q3 & Q1 1b: A)The magnitude of charge Q3 < the magnitude of charge Q1 B) The magnitude of charge Q3 > the magnitude of charge Q1 C) Cannot determine relative magnitudes of charges of Q3 & Q1

8. Q2 Q2 d23 d12 g Q3 Q1 Example 1 A charged ball Q1 is fixed to a horizontal surface as shown. When another massive charged ball Q2 is brought near, it achieves an equilibrium position at a distance d12directly above Q1. When Q1 is replaced by a different charged ball Q3, Q2 achieves an equilibrium position at a distance d23 (< d12) directly above Q3. 1a: A) The charge of Q3 has the same sign of the charge of Q1 B) The charge of Q3 has the opposite sign as the charge of Q1 C) Cannot determine the relative signs of the charges of Q3 & Q1 • To be in equilibrium, the total force on Q2 must be zero. • The only other known force acting on Q2 is its weight. • Therefore, in both cases, the electrical force on Q2 must be directed upward to cancel its weight. • Therefore, the sign of Q3 must be the SAME as the sign of Q1

9. Q2 Q2 d23 d12 g Q3 Q1 Example 1 A charged ball Q1 is fixed to a horizontal surface as shown. When another massive charged ball Q2 is brought near, it achieves an equilibrium position at a distance d12directly above Q1. When Q1 is replaced by a different charged ball Q3, Q2 achieves an equilibrium position at a distance d23 (< d12) directly above Q3. 1b: A) The magnitude of charge Q3 < the magnitude of charge Q1 B) The magnitude of charge Q3 > the magnitude of charge Q1 C) Cannot determine relative magnitudes of charges of Q3 & Q1 • The electrical force on Q2 must be the same in both cases … it just cancels the weight of Q2 • Since d23 < d12 , the charge of Q3 must be SMALLER than the charge of Q1 so that the total electrical force can be the same!!

10. q1 If q1 were the only other charge, we would know the force on q due to q1 - q If q2 were the only other charge, we would know the force on q due to q2 - q2 More Than Two Charges Given charges q, q1, and q2 What is the net force if both charges are present? The net force is given by the Superposition Principle

11. Superposition of Forces • If there are more than two charged objects interacting with each other • The net force on any one of the charged objects is • The vector sum of the individual Coulomb forces on that charged object

12. y (cm) 4 3 2 1 qo q1 q q2 1 2 3 4 5 x(cm) We have that Decompose into its x and y components Example Two qo, q1, and q2 are all point charges where qo = -1mC, q1 = 3mC, and q2 = 4mC • What is the force acting on qo? What are F0xand F0y?

13. y (cm) 4 3 2 1 Now add the components of and to find and qo q1 q2 1 2 3 4 5 x(cm) Example Two - Continued X-direction: Y-direction:

14. y (cm) 4 3 2 1 qo q1 q2 1 2 3 4 5 x(cm) The magnitude of is Example Two - Continued Putting in the numbers . . . We then get for the components At an angle given by

15. Note on constants • k is in reality defined in terms of a more fundamental constant, known as the permittivity of free space.

16. Electric Field • The Electric Force is like the Gravitational Force • Action at a Distance • The electric force can be thought of as being mediated by an electric field.

17. What is a Field? AFieldis something that can be defined anywhere in space • A field represents some physical quantity • (e.g., temperature, wind speed, force) It can be ascalar field(e.g., Temperature field) It can be avector field(e.g., Electric field) It can be a“tensor”field (e.g., Space-time curvature)

18. 72 73 77 75 71 82 77 84 68 80 64 73 83 82 88 55 66 88 80 75 88 90 83 92 91 A Scalar Field A scalar field is a map of a quantity that has only a magnitude, such as temperature

19. 72 73 77 75 71 82 77 84 68 80 64 73 83 56 55 57 66 88 80 75 88 90 83 92 91 A Vector Field A vector field is a map of a quantity that is a vector, a quantity having both magnitude and direction, such as wind

20. Electric Field • We say that when a charged object is put at a point in space, • The charged object sets up an Electric Field throughout the space surrounding the charged object • It is this field that then exerts a force on another charged object

21. Electric Field • Like the electric force, • the electric field is also a vector • If there is an electric force acting on an object having a charge qo, then the electric field at that point is given by (with the sign of q0 included)

22. Electric Field The force on a positively charged object is in the same direction as the electric field at that point, While the force on a negative test charge is in the opposite direction as the electric field at the point

23. Electric Field A positive charge sets up an electric field pointing away from the charge A negative charge sets up an electric field pointing towards the charge

24. Electric Field Earlier we saw that the force on a charged object is given by The term in parentheses remains the same if we change the charge on the object at the point in question The quantity in the parentheses can be thought of as the electric field at the point where the test object is placed The electric field of a point charge can then be shown to be given by

25. Electric Field • As with the electric force, if there are several charged objects, the net electric field at a given point is given by the vector sum of the individual electric fields

26. Electric Field If we have a continuous charge distribution the summation becomes an integral

27. Hints • 1) Look for and exploit symmetries in the problem. • 2) Choose variables for integration carefully. • 3) Check limiting conditions for appropriate result

28. Electric FieldRing of Charge

29. Electric FieldLine of Charge

30. 1) What is the direction of the electric field at point A? a) up b) down c) left d) right e) zero 2) What is the direction of the electric field at point B? a) up b) down c) left d) right e) zero Example 3 Two equal, but opposite charges are placed on the x axis. The positive charge is placed at x = -5 m and the negative charge is placed at x = +5m as shown in the figure above.

31. y E Q1 d Q2 x E E (a) (b) (c) E Q1 Q1 Q2 Q2 Q1 Q2 Example 4 Two charges, Q1 and Q2, fixed along the x-axis as shown produce an electric field, E, at a point (x,y) = (0,d) which is directed along the negative y-axis. Which of the following is true? (a) Both charges Q1 and Q2 are positive (b)Both charges Q1 and Q2 are negative (c)The charges Q1 and Q2 have opposite signs

32. Electric Field Lines • Possible to map out the electric field in a region of space • An imaginary line that at any given point has its tangent being in the direction of the electric field at that point • The spacing, density, of lines is related to the magnitude of the electric field at that point

33. Electric Field Lines • At any given point, there can be only one field line • The electric field has a unique direction at any given point • Electric Field Lines • Begin on Positive Charges • End on Negative Charges

34. Electric Field Lines

35. Electric Dipole • An electric dipole is a pair of point charges having equal magnitude but opposite sign that are separated by a distance d. • Two questions concerning dipoles: • 1) What are the forces and torques acting on a dipole when placed in an external electric field? • 2) What does the electric field of a dipole look like?

36. Force on a Dipole Given a uniform external field Then since the charges are of equal magnitude, the force on each charge has the same value However the forces are in opposite directions! Therefore the net force on the dipole is Fnet = 0

37. Torque on a Dipole • The individual forces acting on the dipole may not necessarily be acting along the same line. • If this is the case, then there will be a torque acting on the dipole, causing the dipole to rotate.

38. Torque on a Dipole The torque is then given by t = qE dsinf d is a vector pointing from the negative charge to the positive charge

39. Potential Energy of a Dipole • Given a dipole in an external field: • Dipole will rotate due to torque • Electric field will do work • The work done is the negative of the change in potential energy of the dipole • The potential energy can be shown to be

40. Electric Field of a Dipole