1. Lesson 3 Forces and Fields

2. Using the Lorentz Force Law

3. Class 7
Today we will:
learn how to do problems using the Lorentz force law.
learn the meaning of the electric potential and how it relates to the potential energy.
learn three ways of representing fields geometrically.

4. Cross products of unit vectors in Cartesian coordinates can be useful You should memorize these.

5. The Lorentz Force Law We showed this was a consequence of
Coulomb’s Law
Special Relativity
Thread Model

6. Using the Lorentz Force Law
In some problems you will be given expressions for the electric and magnetic fields and will be asked to find the force on charges in the field.

7. Using the Lorentz Force Law
In some problems you will be given expressions for the electric and magnetic fields and will be asked to find the force on charges in the field.
A charge of –6.0 μC is placed in an
electric field given by the expression

8. Using the Lorentz Force Law
In some problems you will be given expressions for the electric and magnetic fields and will be asked to find the force on charges in the field.
A charge of –6.0 μC is placed in an
electric field given by the expression
1) What is the direction of the force
on the charge?
Since the charge is negative, the force is
opposite in direction to the electric field.

9. Using the Lorentz Force Law
In some problems you will be given expressions for the electric and magnetic fields and will be asked to find the force on charges in the field.
A charge of –6.0 μC is placed in an
electric field given by the expression
2) Find the force on the charge.

10. Using the Lorentz Force Law
In some problems you will be given expressions for the electric and magnetic fields and will be asked to find the force on charges in the field.
A charge of –6.0 μC is placed in an
magnetic field given by the expression
The charge is at rest.
What is the force on the charge?

11. Using the Lorentz Force Law
In some problems you will be given expressions for the electric and magnetic fields and will be asked to find the force on charges in the field.
A charge of –6.0 μC is placed in an
magnetic field given by the expression
The charge is at rest.
What is the force on the charge?
Since the velocity is zero, there is
no force.

12. Using the Lorentz Force Law
In some problems you will be given expressions for the electric and magnetic fields and will be asked to find the force on charges in the field.
A charge of –6.0 μC is placed in an
magnetic field given by the expression
The velocity of the charge is
What is the direction of the force?

13. Using the Lorentz Force Law
In some problems you will be given expressions for the electric and magnetic fields and will be asked to find the force on charges in the field.
A charge of –6.0 μC is placed in an
magnetic field given by the expression
The velocity of the charge is
What is the direction of the force?
Into the screen is out of the screen, but
q is negative, so the direction is reversed.

14. Using the Lorentz Force Law
In some problems you will be given expressions for the electric and magnetic fields and will be asked to find the force on charges in the field.
A charge of –6.0 μC is placed in an
magnetic field given by the expression
The velocity of the charge is
2) What is the force on the charge?

15. Using the Lorentz Force Law
In some problems you will be given expressions for the electric and magnetic fields and will be asked to find the force on charges in the field.
A charge of –6.0 μC is placed in an
magnetic field given by the expression
The velocity of the charge is
2) What is the force on the charge?

16. Doing the Cross Product Method 1
Draw a coordinate system. Remember that
Then you can find all the cross products of
unit vectors by using the right-hand rule.
For example:

17. Doing the Cross Product Method 1

18. Doing the Cross Product Method 1
Note: Replace the terms one at a time, one step at a time – they’ll end up like the
next slide.

19. Doing the Cross Product Method 1

20. Doing the Cross Product Method 2
Use a determinant:

21. Electric Potential and Potential Energy

22. Electric Field and Electric Potential
The x-component of the force is given by:
Now divide by the field particle’s charge:
We know Fx = qf Ex We let U = qf V
V is called the “electric potential” or “voltage.”

23. Electric Field and Electric Potential
force is to potential energy as
electric field is to electric potential

24. Electric Field and Electric Potential
Electric potential is used with static charges.
We think of charges producing fields throughout space. The electric field is a vector field that gives the force on test charges:
The electric potential is a scalar field that gives the potential energy of test charges:

25. Electric Field and Electric Potential
Electric potential is also used in circuits.
In circuits, electric potential is usually called voltage.
Voltage is (the change in potential energy of an electron between two points in a circuit)/(electron charge)
To understand of voltage, always think of an electron’s potential energy.

26. Electric Field and Electric Potential
The electric field and electric potential are related by equations similar to those for the force and the potential energy:

27. Electric Field and Electric Potential
What we call zero electric potential is arbitrary, just as the zero of potential energy is arbitrary.
We usually say charges have zero electric potential when they are far apart.
We usually say that circuits have zero voltage at points where they are “grounded.” (Ground is zero volts.)

28. Electric Field and Electric Potential
For point charges we have:

29. Ways of Representing Electric and Magnetic Fields

30. 1. Field Vectors
At every point in space there is a field vector defined.
The fields satisfy the Lorentz Force Law.
We can find the field vectors by:
Theory
Computation
Experiment

31. 1. Field Vectors We usually represent the field vectors by equations.
Sometimes we represent fields by collections of numbers on a 3-D mesh (3 numbers per point)

32. 1. Field Vectors We can also draw field vectors.
Magnitude: length of arrow
Direction: direction of arrow

33. 2. Field Lines We can make field lines by
Aligning threads (E field) or stubs (B field)
Start at one point, find the field, move along the direction of field to a nearby point, find the field, …

34. Electric Field Lines
Consider a positive charge and a negative charge.

35. Electric Field Lines
Take one vector pointing in the direction of the electric
field near a positive charge.

36. Electric Field Lines
Then attach the head of a second vector to the tail of the
first vector.

37. Electric Field Lines
Continue this process over and over.

38. Electric Field Lines
This line is an electric field line.

39. Electric Field Lines
The direction of the electric field is tangent to the electric field line

40. Electric Field Lines
The electric field is stronger where the lines are closer together.
weaker field
stronger field

41. Electric Field Lines
The electric field is stronger where the lines are closer together.
weaker field
stronger field

42. Field Lines, A Summary:
Electric field lines go from positive charges toward negative charges.
The direction of the electric field at a point is tangent to the field line.
The field is stronger where field lines are closer together.
We’ll learn a lo more about field lines in Lesson 5.

43. 2. Field Lines
Align some threads in a box with the ends of the box perpendicular to the threads. The height of the box is ℓ. A ℓ

44. 2. Field Lines
Align some threads in a box with the ends of the box perpendicular to the threads. The height of the box is ℓ.
Each thread is part of one field line when aligned. A ℓ

45. 2. Field Lines
Align some threads in a box with the ends of the box perpendicular to the threads. The height of the box is ℓ.
Each thread is part of one field line when aligned.
The force is proportional to the density of threads time the length of the threads. A ℓ

46. 2. Field Lines
Align some threads in a box with the ends of the box perpendicular to the threads. The height of the box is ℓ.
Each thread is part of one field line when aligned.
The force is proportional to the number of field lines per unit area. A ℓ

47. 2. Field Lines
Align some threads in a box with the ends of the box perpendicular to the threads. The height of the box is ℓ.
Each thread is part of one field line when aligned.
The force is proportional to the number of field lines per unit area – through a surface ┴ to the lines. A ℓ

48. 2. Field Lines When we draw field lines:
Magnitude: strong where lines are close together
Direction: tangent to field lines

49. 3. Field Contours We make a field contour by
forming a continuous surface that is always perpendicular to the field.
Taking a set of perpendicular surfaces with the spacing between surfaces closer where the field is stronger.

50. 3. Field Contours
For static electric fields, the perpendicular surfaces are surfaces of constant potential energy.
These surfaces are also called “equipotential surfaces.”

51. 3. Field Contours When we draw field contours:
Magnitude: strong where surfaces are close together
Direction: normal to field contours

52. In and Out of the Board
Think of arrows coming out of the screen. If you see the shaft, the arrow is coming out of the screen. If you see the fletching, the arrow is going into the screen.
Out In

53. Conductors and Fields

54. Static Electric Fields in a Conductor
If there is a net electric field inside a conductor, no matter how small, charges will move.
Animate the following. Start with equal numbers of + and - charges scattered throughout.

55. Static Electric Fields in a Conductor
If there is a net electric field inside a conductor, no matter how small, charges will move.
Apply an external field, the – charge move opposite the field arrows.

56. Static Electric Fields in a Conductor
If there is a net electric field inside a conductor, no matter how small, charges will move.
Apply an external field, the – charge move a little opposite the field arrows. The number of – charges and + charges inside should still be equal, but there should be net negative charge on the left and net + charge on the right.

57. Static Electric Fields in a Conductor
If there is a net electric field inside a conductor, no matter how small, charges will move.
Inside a conductor, there can be no static electric fields. (E=0 inside)
On the surface of a conductor, the static electric field must have no component parallel to the surface. (E is perpendicular to the surface)

58. Static Electric Fields in a Conductor
A conductor is near a point charge. In (a) we see only the field of the point charge. In (b) we see only the field of the conductor.
Charges rearrange themselves in the conductor so
No net charge is on the inside of the conductor.
Surface charge completely cancels out E inside.
The net field has E=0 inside, and E perpendicular to the surface.
(b)
(a)
(c)
So what do the perpendicular surfaces – equipotential surfaces look like?

59. Equipotential Surfaces
Since the static electric field is perpendicular to the surface of a conductor, the surface is an element of a field contour and an equipotential surface.
The electric potential is a constant throughout a conductor as well as on the surface.
So what do the perpendicular surfaces – equipotential surfaces look like?
(c)

60. Class 8
Today we will:
learn about static electric fields and electric potentials in conductors.
learn how charges move in regions where there are electric and magnetic fields.
learn about some practical devices that use electric and magnetic fields.

61. The Motion of Charged Particles in Electric and Magnetic Fields

62. Motion of Charges in a Constant Electric Field
Constant force just like with gravity.
Acceleration is

63. Motion of Charges in a Constant Electric Field+ +
Animate this… The + charge moves right with increasing speed.
+ charge from rest

64. Motion of Charges in a Constant Electric Field+ +
+ charge from rest

65. Motion of Charges in a Constant Electric Field+ +
Animate again – the – charge moves left eith increasing speed.
− charge from rest

66. Motion of Charges in a Constant Electric Field+ +
− charge from rest

67. Motion of Charges in a Constant Electric Field+ +
− charge with initial velocity

68. Motion of Charges in a Constant Electric Field+ +
The upward speed remains constant while the sideways speed slows down and
turns around. Leave a trace of the particle’s path.
+ charge with initial velocity

69. Forces in a Constant Electric Field
0 V
+200 V
+ charge at rest
q = 1.50 mC
2.00cm

70. Forces in a Constant Electric Field
0 V
+200 V
+ charge at rest
q = 1.50 mC
2.00cm

71. Forces in a Constant Electric Field
0 V
+200 V
+ charge at rest
q = 1.50 mC
2.00cm

72. Forces in a Constant Electric Field
0 V
+200 V
+ charge at rest
q = 1.50 mC
2.00cm

73. Kinetic and Potential Energy in a Constant Electric Field
0 V
+200 V
+ charge at rest
q = 1.50 mC
2.00cm

74. Kinetic and Potential Energy in a Constant Electric Field
For a charge starting at rest and accelerated through an electric potential ΔV:

75. Motion of Charges in a Constant Magnetic Field
This is just weird…
The force is perpendicular to the velocity, so it can only change the direction of motion without speeding up or slowing down the charge.

76. Motion of Charges in a Constant Magnetic Field – v Parallel to B
If the charge moves parallel (or antiparallel) to B, there is no force at all.

77. Motion of Charges in a Constant Magnetic Field – v Perpendicular to B
The direction of the force is given by the right-hand rule. It is in a plane perpendicular to B and is at 90° to v.
This force keeps the charge moving in a circle.

78. Motion of Charges in a Constant Magnetic Field – v Perpendicular to B
Animate the charge moving in a circle at constant angular speed. v is always
tangent and F is always toward the center.

79. Motion of Charges in a Constant Magnetic Field – v Perpendicular to B

80. Motion of Charges in a Constant Magnetic Field – v Perpendicular to B

81. What happens when there is a component of v parallel to B and a component perpendicular to B?

82. Motion of Charges in a Constant Magnetic Field – v General
This is a helical path with the charge moving out of the screen with a constant
speed in that direction. Can you animate it in two-dimensions (where the motion
is still circular), and then rotate the view to see the helical motion?

83. Two Useful Equations Know how to derive these!
Cyclotron Radius
This even works for relativistic particles
as long as we use the relativistic momentum.

84. Two Useful Equations Know how to derive these!
Cyclotron Frequency
This is good relativistically if we use the relativistic mass.

85. Devices Using Electric and Magnetic Fields

86. Linear Accelerators Series of drift tubes.
E inside drift tubes is very small.
Change V at a high frequency so it always pushes the particle forward.
Drift tubes get longer as v increases.
+V −V V
This would be fun if it could be animated. – The speed remains constant when
the proton is within one section, but it speeds up in between sections….

87. Cyclotrons Dees instead of drift tubes.
Path in dees naturally gets longer.
This would also be a nice animation. The proton moves in a circular path
at constant speed within each dee, but speeds up each time it passes from
one dee to the next.

88. Mass Spectrometers Accelerate ions in electric field.
Bend ions in magnetic field.
Here the positive ion speeds up in the E region, then maintains the
same speed while moving in a circular path in the B region.
It would be nice to show two ions side by side, one more massive
than the other. The more massive ion doesn’t go as fast and then
bends with a smaller radius in the B region.
One version….