The magnitude of the force

1. The magnitude of the force Two equal charges Q are placed a certain distance apart. They exert equal-and-opposite forces F on one another. Now one of the charges is doubled in magnitude to 2Q. What happens to the magnitude of the force each charge experiences? 1. Both charges experience forces of magnitude 2F. 2. The Q charge experiences a force of 2F; the 2Q charge experiences a force F. 3. The Q charge experiences a force of F; the 2Q charge experiences a force 2F. 4. None of the above.

2. The magnitude of the force Let’s examine this question from two perspectives. Newton’s Third Law – can one object experience a larger force than the other? 2. Coulomb’s Law – if we double one charge, what happens to the force?

3. The magnitude of the force Let’s examine this question from two perspectives. Newton’s Third Law – can one object experience a larger force than the other? No – the objects experience equal-and-opposite forces. 2. Coulomb’s Law – if we double one charge, what happens to the force?

4. Superposition If an object experiences multiple forces, we can use: The principle of superposition - the net force acting on an object is the vector sum of the individual forces acting on that object.

5. Worksheet – a one-dimensional situation Ball A, with a mass 4m, is placed on the x-axis at x = 0. Ball B, which has a mass m, is placed on the x-axis at x = +4a. Where would you place ball C, which also has a mass m, so that ball A feels no net force because of the other balls? Is this even possible?

6. Worksheet – a one-dimensional situation Ball A, with a mass 4m, is placed on the x-axis at x = 0. Ball B, which has a mass m, is placed on the x-axis at x = +4a. Where would you place ball C, which also has a mass m, so that ball A feels no net force because of the other balls? Is this even possible?

7. Worksheet – a one-dimensional situation Ball A, with a mass 4m, is placed on the x-axis at x = 0. Ball B, which has a mass m, is placed on the x-axis at x = +4a. Could you re-position ball C, which also has a mass m, so that ball B feels no net force because of the other balls?

8. Worksheet – a one-dimensional situation Ball A, with a mass 4m, is placed on the x-axis at x = 0. Ball B, which has a mass m, is placed on the x-axis at x = +4a. Could you re-position ball C, which also has a mass m, so that ball B feels no net force because of the other balls?

9. Worksheet – a one-dimensional situation Ball A, with a charge +4q, is placed on the x-axis at x = 0. Ball B, which has a charge –q, is placed on the x-axis at x = +4a. Where would you place ball C, which has a charge of magnitude q, and could be positive or negative, so that ball A feels no net force because of the other balls?

10. Worksheet – a one-dimensional situation Ball A, with a charge +4q, is placed on the x-axis at x = 0. Ball B, which has a charge –q, is placed on the x-axis at x = +4a. Where would you place ball C, which has a charge of magnitude q, and could be positive or negative, so that ball A feels no net force because of the other balls?

11. Worksheet – a one-dimensional situation Ball A, with a charge +4q, is placed on the x-axis at x = 0. Ball B, which has a charge –q, is placed on the x-axis at x = +4a. Where would you place ball C, which has a charge of magnitude q, and could be positive or negative, so that ball A feels no net force because of the other balls?

12. Worksheet – a one-dimensional situation Ball A, with a charge +4q, is placed on the x-axis at x = 0. Ball B, which has a charge –q, is placed on the x-axis at x = +4a. Could you re-position ball C, which has a charge of magnitude q, and could be positive or negative, so that ball B is the one feeling no net force?

13. Worksheet – a one-dimensional situation Ball A, with a charge +4q, is placed on the x-axis at x = 0. Ball B, which has a charge –q, is placed on the x-axis at x = +4a. Could you re-position ball C, which has a charge of magnitude q, and could be positive or negative, so that ball B is the one feeling no net force?

14. Worksheet – a one-dimensional situation Ball A, with a charge +4q, is placed on the x-axis at x = 0. Ball B, which has a charge –q, is placed on the x-axis at x = +4a. Could you re-position ball C, which has a charge of magnitude q, and could be positive or negative, so that ball B is the one feeling no net force?

15. A two-dimensional situation Simulation Case 1: There is an object with a charge of +Q at the center of a square. Can you place a charged object at each corner of the square so the net force acting on the charge in the center is directed toward the top right corner of the square? Each charge has a magnitude of Q, but you get to choose whether it is + or – .

16. Case 1 – let me count the ways. There is an object with a charge of +Q at the center of a square. Can you place a charged object at each corner of the square so the net force acting on the charge in the center is directed toward the top right corner of the square? Each charge has a magnitude of Q, but you get to choose whether it is + or – . How many possible configurations can you come up with that will produce the required force? 1. 1 2. 2 3. 3 4. 4 5. either 0 or more than 4

17. A two-dimensional situation Simulation Case 2: The net force on the positive center charge is straight down. What are the signs of the equal-magnitude charges occupying each corner? How many possible configurations can you come up with that will produce the desired force?

18. Case 2 – let me count the ways. There is an object with a charge of +Q at the center of a square. Can you place a charged object at each corner of the square so the net force acting on the charge in the center is directed straight down? Each charge has a magnitude of Q, but you get to choose whether it is + or – . How many possible configurations can you come up with that will produce the required force? 1. 1 2. 2 3. 3 4. 4 5. either 0 or more than 4

19. A two-dimensional situation Simulation Case 3: There is no net net force on the positive charge in the center. What are the signs of the equal-magnitude charges occupying each corner? How many possible configurations can you come up with that will produce no net force?

20. Case 3 – let me count the ways. There is an object with a charge of +Q at the center of a square. Can you place a charged object at each corner of the square so there is no net force acting on the charge in the center? Each charge has a magnitude of Q, but you get to choose whether it is + or – . How many possible configurations can you come up with that will produce no net force? 1. 1 2. 2 3. 3 4. 4 5. either 0 or more than 4

21. Worksheet: a 1-dimensional example Three charges are equally spaced along a line. The distance between neighboring charges is a. From left to right, the charges are: q1 = –Qq2 = +Qq3 = +Q What is the magnitude of the force experienced by q2, the charge in the center? Simulation

22. Worksheet: a 1-dimensional example Let's define positive to the right. The net force on q2 is the vector sum of the forces from q1 and q3. The force has a magnitude of and points to the left. Handling the signs correctly is critical. The negative signs come from the direction of each of the forces (both to the left), not from the signs of the charges. I generally drop the signs in the equation and get any signs off the diagram by drawing in the forces.

23. Ranking based on net force • Rank the charges according to the magnitude of the net force they experience, from largest to smallest. • 1. 1 = 2 > 3 • 2. 1 > 2 > 3 • 3. 2 > 1 = 3 • 4. 2 > 1 > 3 • 5. None of the above.

24. Ranking based on net force Will charges 1 and 3 experience forces of the same magnitude? Will charges 1 and 2 experience forces of the same magnitude (both have two forces acting in the same direction)?

25. Ranking based on net force Will charges 1 and 3 experience forces of the same magnitude? No, because both forces acting on charge 1 are in the same direction, while the two forces acting on charge 3 are in opposite directions. Thus, 1 > 3. Will charges 1 and 2 experience forces of the same magnitude (both have two forces acting in the same direction)?

26. Ranking based on net force Will charges 1 and 3 experience forces of the same magnitude? No, because both forces acting on charge 1 are in the same direction, while the two forces acting on charge 3 are in opposite directions. Thus, 1 > 3. Will charges 1 and 2 experience forces of the same magnitude (both have two forces acting in the same direction)? No, because one force acting on charge 1 is the same magnitude as one acing on charge 2, while the second force acting on charge 1 is smaller – it comes from a charge farther away. Thus, 2 > 1.

27. Ranking based on net force We can calculate the net force, too. If we add these forces up, what do we get? Is that a fluke?

28. Three charges in a line • Ball 1 has an unknown charge and sign. Ball 2 is positive, with a charge of +Q. Ball 3 has an unknown non-zero charge and sign. • Ball 3 is in equilibrium - it feels no net electrostatic force due to the other two balls. • What is the sign of the charge on ball 1? • 1. Positive • 2. Negative • 3. We can't tell unless we know the sign of the charge on ball 3.

29. Three charges in a line Ball 3 is in equilibrium because it experiences equal-and-opposite forces from the other two balls, so ball 1 must have a negative charge. Flipping the sign of the charge on ball 3 reverses both these forces, so they still cancel.

30. Three charges in a line What is the magnitude of the charge on ball 1? Can we even tell if we don’t know what Q3 is?

31. Three charges in a line What is the magnitude of the charge on ball 1? Can we even tell if we don’t know what Q3 is? Yes, we can! For the two forces to be equal-and-opposite, with ball 1 three times as far from ball 3 as ball 2 is, and the distance being squared in the force equation, the charge on ball 1 must have a magnitude of 9Q.

32. Three charges in a line Let’s do the math. Define to the right as positive.

33. Two charges in a line The neat thing here is that we don't need to know anything about ball 3. We can put whatever charge we like at the location of ball 3 and it will feel no net force because of balls 1 and 2. Ball 3 isn't special - it's the location that's special. So, let's get rid of ball 3 from the picture and think about how the two charged balls influence the point where ball 3 was.

34. Two charges in a line Ball 2's effect on ball 3 is given by Coulomb's Law: Ball 2's effect on the point where ball 3 was is given by Electric Field :   The electric field from ball 1 and the electric field from ball 2 cancel out at the location where ball 3 was.

35. Electric field A field is something that has a magnitude and a direction at every point in space. An example is a gravitational field, symbolized by g. The electric field, E, plays a similar role for charged objects that g does for objects that have mass. g has a dual role, because it is also the acceleration due to gravity. If only gravity acts on an object: For a charged object acted on by an electric field only, the acceleration is given by: Simulation

36. Electric field lines Field line diagrams show the direction of the field, and give a qualitative view of the magnitude of the field at various points. The field is strongest where the lines are closer together. a – a uniform electric field directed down b – the field near a negative point charge c – field lines start on positive charges and end on negative charges. This is an electric dipole – two charges of opposite sign and equal magnitude separated by some distance.

37. Electric field vectors Field vectors give an alternate picture, and reinforce the idea that there is an electric field everywhere. The field is strongest where the vectors are darker. a – a uniform electric field directed down b – the field near a negative point charge c – field lines start on positive charges and end on negative charges. This is an electric dipole – two charges of opposite sign and equal magnitude separated by some distance.

38. Getting quantitative about field The field line and field vector diagrams are nice, but when we want to know about the electric field at a particular point those diagrams are not terribly useful. Instead, we use superposition. The net electric field at a particular point is the vector sum of the individual electric fields at that point. The individual fields sometimes come from individual charges. We assume these charges to be highly localized, so we call them point charges. Electric field from a point charge: The field points away from a + charge, and towards a – charge.

39. A triangle of point charges Three point charges, having charges of equal magnitude, are placed at the corners of an equilateral triangle. The charge at the top vertex is negative, while the other two are positive. In what direction is the net electric field at point A, halfway between the positive charges? We could ask the same question in terms of force. In what direction is the net electric force on a ______ charge located at point A?

40. A triangle of point charges Three point charges, having charges of equal magnitude, are placed at the corners of an equilateral triangle. The charge at the top vertex is negative, while the other two are positive. In what direction is the net electric field at point A, halfway between the positive charges? We could ask the same question in terms of force. In what direction is the net electric force on a positive charge located at point A?

41. Net electric field at point A • In what direction is the net electric field at point A, halfway between the positive charges? • 1. up • 2. down • 3. left • 4. right • 5. other

42. Net electric field at point A The fields from the two positive charges cancel one another at point A. The net field at A is due only to the negative charge, which points toward the negative charge (up).

43. Are there any locations, a finite distance from the charges, on the straight line passing through point A and the negative charge at which the net electric field due to the charges equals zero? If so, where is the field zero? 1. At some point above the negative charge 2. At some point between the negative charge and point A 3. At some point below point A 4. Both 1 and 3 5. Both 2 and 3 6. None of the above Net electric field equals zero?

44. Net electric field equals zero? Simulation Inside the triangle, the field from the negative charge is directed up. What about the fields from the two positive charges? Do they have components up or down? At the top, and at point A, the field is dominated by ____________. Far away, the field is dominated by ______________.

45. Net electric field equals zero? Simulation Inside the triangle, the field from the negative charge is directed up. What about the fields from the two positive charges? Do they have components up or down? Up. Thus, the net field everywhere inside the triangle has a component up. At the top, and at point A, the field is dominated by the negative charge. Far away, the field is dominated by the positive charges. In between, there must be a balance.

46. Worksheet: where is the field zero? Two charges, +3Q and –Q, are separated by 4 cm. Is there a point along the line passing through them (and a finite distance from the charges) where the net electric field is zero? If so, where? First, think qualitatively. Is there such a point to the left of the +3Q charge? Between the two charges? To the right of the –Q charge?

47. Where is the net field equal to zero? Is the net electric field equal to zero at some point in one of these three regions: to the left of both charges (Region I), in between both charges (Region II), and/or to the right of both charges (Region III)? The field is zero at a point in: 1. Region I 2. Region II 3. Region III 4. two of the above 5. all of the above

48. Worksheet: where is the field zero? In region I, the two fields point in opposite directions. In region II, both fields are directed to the right, so they cannot cancel. In region III, the two fields point in opposite directions. Now think about the magnitude of the fields.

49. Worksheet: where is the field zero? In region II, both fields are directed to the right, so they cannot cancel. In region I, every point is closer to the larger-magnitude charge than the smaller-magnitude charge, so the field from the +3Q charge will always be larger than that from the –Q charge.

50. Worksheet: where is the field zero? In region II, both fields are directed to the right, so they cannot cancel. In region I, the fields cannot cancel, either. In region III, we can strike a balance between the factor of 3 in the charges and the distances.