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

• Lets examine this question from two
  perspectives.
• Newtons Third Law can one object experience a
  larger force than the other?
• 2. Coulombs Law if we double one charge, what
  happens to the force?

3
The magnitude of the force

• Lets examine this question from two
  perspectives.
• Newtons Third Law can one object experience a
  larger force than the other?
• No the objects experience equal-and-opposite
  forces.
• 2. Coulombs 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 Q q2 Q q3 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 gt 3
• 2. 1 gt 2 gt 3
• 3. 2 gt 1 3
• 4. 2 gt 1 gt 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 gt
  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 gt
  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 gt 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 dont 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 dont 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

• Lets 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
Net electric field equals zero?
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
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?
• Simulation

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 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.
• 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.
• In region III, we can strike a balance between
  the factor of 3 in the charges and the distances.

50
Worksheet where is the field zero?

• How can we calculate where the point is? If the
  point is a distance x from the 3Q charge, then
  it is (x 4) cm away from the -Q charge. Define
  right as positive, so
• The minus sign in front of the second term is not
  the one associated with the charge but the one
  associated with the direction of the field from
  the charge.
• The k's and Q's cancel. Re-arranging gives
• Cross multiplying and expanding the brackets

51
Worksheet where is the field zero?

• Solve this using the quadratic formula to get two
  solutions
• The two solutions are x 2.54 cm and x 9.46
  cm. Which one is correct?

52
Two solutions
Which of the two solutions is the one we want?
1. 2.54 cm 2. 9.46 cm 3. They are both valid
solutions. Note even if you decide one
solution is not valid, you should be able to
explain what its physical significance is.
53
Where is the field zero?

• The net electric field is zero 9.46 cm to the
  right of the 3Q charge (and 5.46 cm to the right
  of the Q charge).
• The other solution is between the two charges,
  where the two fields point in the same direction.
  This point, 2.54 cm to the right of the 3Q
  charge, is where the two fields are equal.

54
A test charge

• A test charge has a small enough charge that it
  has a negligible impact on the local electric
  field.
• Placing a positive test charge at a point can
  tell us the direction of the electric field at
  that point, and tell us roughly how strong the
  field is.
• The force on a positive test charge is in the
  same direction as the electric field, because
  .
• Simulation

55
The net force on a test charge

• The diagram shows the net force experienced by a
  positive test charge located at the center of the
  diagram. The force comes from two nearby charged
  balls, one with a charge of Q and one with an
  unknown charge. What is the sign and magnitude of
  the charge on the second ball?
• Q/4
• Q/2
• Q
• 2Q
• 4Q
• none of these

56
The net force on a test charge

• This is the same as asking If the net electric
  field at the point at the center of the diagram
  is in the direction shown, what is the sign and
  magnitude of the charge on the second ball?
• The vector is at a 45 angle, so the
• two forces (or fields) must be
• identical. The Q charge sets up a
• force (or field) directed down. The
• second ball must set up a force (or
• field) directed left, away from itself,
• so it must be positive.

57
The net force on a test charge

• If the two forces (or fields) are the same, how
  does the magnitude of the charge on the second
  ball compare to Q?

58
The net force on a test charge

• If the two forces (or fields) are the same, how
  does the magnitude of the charge on the second
  ball compare to Q?
• It must be smaller than Q, because the
• second ball is closer to the point
• were interested in.
• The first ball is twice as far away.
• Because distance is squared in the
• equation, the factor of 2 becomes
• a factor of 4. To offset this factor of
• 4, the second ball has a charge of Q/4.

59
The net force on a test charge, II

• The diagram shows the net force experienced by a
  positive test charge located at the center of the
  diagram. The force comes from two nearby charged
  balls, one with a charge of Q and one with an
  unknown charge. What is the sign and magnitude of
  the charge on the second ball?
• Q
• Q v2
• 2Q
• 2Q v2
• 4Q
• none of these

60
The net force on a test charge, II

• In which direction is the force (or field) from
  the Q charge?
• What are the possible directions for the force
  (or field) from the second ball?

61
The net force on a test charge, II

• In which direction is the force (or field) from
  the Q charge?
• Down, away from the Q charge.
• What are the possible directions for the force
  (or field) from the second ball?
• Left, if it is positive, or right, if it
• is negative.
• Can we combine a vector down
• with a vector left or right to get
• the vector shown?

62
The net force on a test charge, II

• In which direction is the force (or field) from
  the Q charge?
• Down, away from the Q charge.
• What are the possible directions for the force
  (or field) from the second ball?
• Left, if it is positive, or right, if it
• is negative.
• Can we combine a vector down
• with a vector left or right to get
• the vector shown?
• No this situation is not possible.

63
Electric field near conductors, at equilibrium

• A conductor is in electrostatic equilibrium when
  there is no net flow of charge. Equilibrium is
  reached in a very short time after being exposed
  to an external field. At equilibrium, the charge
  and electric field follow these guidelines
• the electric field is zero within the solid part
  of the conductor
• the electric field at the surface of the
  conductor is perpendicular to the surface
• any excess charge lies only at the surface of
  the conductor
• charge accumulates, and the field is strongest,
  on pointy parts of the conductor

64
Electric field near conductors, at equilibrium
At equilibrium the field is zero inside a
conductor and perpendicular to the surface of the
conductor because the electrons in the conductor
move around until this happens. Excess
charge, if the conductor has a net charge, can
only be found at the surface. If any was in the
bulk, there would be a net field inside the
conductor, making electrons move. Usually the
excess charge is on the outer surface.
65
Electric field near conductors, at equilibrium
Charge piles up (and the field is strongest) at
pointy ends of a conductor to balance forces on
the charges. On a sphere, a uniform charge
distribution at the surface balances the forces,
as in (a) below. For charges in a line, a
uniform distribution (b) does not correspond to
equilibrium. Start out with the charges equally
spaced, and the forces the charges experience
push them so that they accumulate at the ends
(c).
66
A lightning rod

• A van de Graaff generator acts like a
  thundercloud. We will place a large metal sphere
  near the van de Graaff and see what kind of
  sparks (lightning) we get. We will then replace
  the large metal sphere by a pointy piece of
  metal. In which case do we get more impressive
  sparks (lightning bolts)?
• with the large sphere
• with the pointy object
• neither, the sparks are the same in the two
  cases

67
A lightning rod

• The big sparks we get with the sphere are
  dangerous, and in real life could set your house
  on fire.
• With the lightning rod, the charge (and field)
  builds up so quickly that it drains charge out of
  the cloud slowly and continuously, avoiding the
  dangerous sparks.
• The lightning rod was invented by __________.

68
A lightning rod

• The big sparks we get with the sphere are
  dangerous, and in real life could set your house
  on fire.
• With the lightning rod, the charge (and field)
  builds up so quickly that it drains charge out of
  the cloud slowly and continuously, avoiding the
  dangerous sparks.
• The lightning rod was invented by Ben Franklin.

69
Electric potential energy (uniform field)

• For an object with mass in a uniform
  gravitational field, the change in gravitational
  potential energy is
• Similarly, for a charge q moving a distance d
  parallel to the electric field, the change in
  electric potential energy is

70
Which way does it go?
Whether it's an object with mass in a
gravitational field, or a charged object in an
electric field, when the object is released from
rest it will accelerate in what direction? 1.
Toward U 0 2. Away from U 0 3. In the
direction of the field 4. In the direction of
decreasing potential energy 5. In the direction
of increasing potential energy
71
Which way does it go?

• Masses and positive charges behave in a similar
  way, but negative charges move opposite in
  direction to positive charges. In all cases, the
  object accelerates in the direction of decreasing
  potential energy. This is true whether the field
  is uniform or non-uniform.
• Simulation

72
Electric potential energy (for point charges)

• There is an electric potential energy associated
  with two charged objects, of charge q and Q,
  separated by a distance r. Note that the
  potential energy is defined to be zero when
• r infinity.
• Potential energy is a scalar, so we handle signs
  differently than we do when we are handling
  vectors. Put the signs on the charges into the
  equation!
• This should remind you of the equivalent
  gravitational situation, in which

Electric potential energy
73
Interacting point charges
Case 1 a charge q is placed at a point near a
large fixed charge Q. Case 2 the q charge is
replaced by a q charge of the same mass. In
which case is the potential energy larger? 1.
Case 1 2. Case 2 3. neither, the potential
energy is equal in both cases
74
Interacting point charges

• In case 1, the potential energy is positive.
• In case 2, the potential energy is negative.
• A positive scalar is bigger than a negative
  scalar (check with your bank manager about your
  bank balance if you have trouble with this
  concept!).
• Simulation

Electric potential energy
75
Interacting point charges
We now release the charges from rest and observe
them for a particular time interval. Assuming no
collisions have taken place, at the end of that
time interval which charge will have the greatest
speed? 1. The q charge 2. The q charge 3.
Both charges will have the same speed
76
Interacting point charges

• In this case, we can apply impulse momentum
  ideas. The negative charge keeps getting closer
  to the central positive charge, so the force it
  feels increases. The opposite happens for the q
  charge. Because the q charge experiences a
  larger average force, its speed is larger after a
  given time interval.

77
Escape speed

• How fast would you have to throw an object so it
  never came back down? Ignore air resistance.
  Let's find the escape speed - the minimum speed
  required to escape from a planet's gravitational
  pull.
• How should we try to figure this out?
• Attack the problem from a force perspective?
• From an energy perspective?

78
Escape speed

• How fast would you have to throw an object so it
  never came back down? Ignore air resistance.
  Let's find the escape speed - the minimum speed
  required to escape from a planet's gravitational
  pull.
• How should we try to figure this out?
• Attack the problem from a force perspective?
• From an energy perspective?
• Forces are hard to work with here, because the
  size of the force changes as the object gets
  farther away. Energy is easier to work with in
  this case.

79
Escape speed

• Lets do an equivalent problem for two charged
  objects.
• Find an expression for the minimum speed an
  electron, which starts some distance r from a
  proton, must have to escape from the proton.
  Assume the proton remains at rest the whole time.
  
• Lets start with the conservation of energy
  equation.
• Which terms can we cross out immediately?

80
Escape speed

• Which terms can we cross out immediately?
• Assume no resistive forces, so
• Assume the electron barely makes it
• to infinity, so both Uf and Kf are zero.
• This leaves

81
Escape speed

• If the total mechanical energy is negative, the
  object comes back. If it is positive, it never
  comes back.
• Solving for the escape speed gives
• m is the mass of the electron r is the initial
  distance between them. For an electron in the
  hydrogen ground state, we get vescape 3.1 106
  m/s.

82
Releasing two charges

• Simulation
• Two charged objects are placed close to one
  another and released from rest. Assume that each
  object is affected only by the other object.

83
Releasing two charges
In the first case, we observe that the motion of
one object is a mirror image of the motion of the
other. What, if anything, can we say about the
two objects? 1. They have the same mass. 2.
They have the same charge (sign and magnitude).
3. Both of the above. 4. Neither of the above
has to be true.
84
Releasing two charges

• How do the accelerations compare?
• How do the forces compare? (Can you answer this
  if you dont know how the charges compare?)
• How do the masses compare?

85
Releasing two charges

• How do the accelerations compare?
• The accelerations are equal-and-opposite.
• How do the forces compare? (Can you answer this
  if you dont know how the charges compare?)
• The forces are equal-and-opposite, even if the
  charges are different (Newtons Third Law).
• How do the masses compare?
• They are the same, because m F/a.