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Physics

18.3 Electric Field

Physics18.3 Electric Field
  1. Preface
  2. 1 What is Physics?
    1. Introduction
    2. 1.1 Physics: Definitions and Applications
    3. 1.2 The Scientific Methods
    4. 1.3 The Language of Physics: Physical Quantities and Units
    5. Key Terms
    6. Section Summary
    7. Key Equations
    8. Chapter Review
      1. Concept Items
      2. Critical Thinking Items
      3. Problems
      4. Performance Task
    9. Test Prep
      1. Multiple Choice
      2. Short Answer
      3. Extended Response
  3. 2 Motion in One Dimension
    1. Introduction
    2. 2.1 Relative Motion, Distance, and Displacement
    3. 2.2 Speed and Velocity
    4. 2.3 Position vs. Time Graphs
    5. 2.4 Velocity vs. Time Graphs
    6. Key Terms
    7. Section Summary
    8. Key Equations
    9. Chapter Review
      1. Concept Items
      2. Critical Thinking Items
      3. Problems
      4. Performance Task
    10. Test Prep
      1. Multiple Choice
      2. Short Answer
      3. Extended Response
  4. 3 Acceleration
    1. Introduction
    2. 3.1 Acceleration
    3. 3.2 Representing Acceleration with Equations and Graphs
    4. Key Terms
    5. Section Summary
    6. Key Equations
    7. Chapter Review
      1. Concept Items
      2. Critical Thinking Items
      3. Problems
      4. Performance Task
    8. Test Prep
      1. Multiple Choice
      2. Short Answer
      3. Extended Response
  5. 4 Forces and Newton’s Laws of Motion
    1. Introduction
    2. 4.1 Force
    3. 4.2 Newton's First Law of Motion: Inertia
    4. 4.3 Newton's Second Law of Motion
    5. 4.4 Newton's Third Law of Motion
    6. Key Terms
    7. Section Summary
    8. Key Equations
    9. Chapter Review
      1. Concept Items
      2. Critical Thinking Items
      3. Problems
      4. Performance Task
    10. Test Prep
      1. Multiple Choice
      2. Short Answer
      3. Extended Response
  6. 5 Motion in Two Dimensions
    1. Introduction
    2. 5.1 Vector Addition and Subtraction: Graphical Methods
    3. 5.2 Vector Addition and Subtraction: Analytical Methods
    4. 5.3 Projectile Motion
    5. 5.4 Inclined Planes
    6. 5.5 Simple Harmonic Motion
    7. Key Terms
    8. Section Summary
    9. Key Equations
    10. Chapter Review
      1. Concept Items
      2. Critical Thinking Items
      3. Problems
      4. Performance Task
    11. Test Prep
      1. Multiple Choice
      2. Short Answer
      3. Extended Response
  7. 6 Circular and Rotational Motion
    1. Introduction
    2. 6.1 Angle of Rotation and Angular Velocity
    3. 6.2 Uniform Circular Motion
    4. 6.3 Rotational Motion
    5. Key Terms
    6. Section Summary
    7. Key Equations
    8. Chapter Review
      1. Concept Items
      2. Critical Thinking Items
      3. Problems
      4. Performance Task
    9. Test Prep
      1. Multiple Choice
      2. Short Answer
      3. Extended Response
  8. 7 Newton's Law of Gravitation
    1. Introduction
    2. 7.1 Kepler's Laws of Planetary Motion
    3. 7.2 Newton's Law of Universal Gravitation and Einstein's Theory of General Relativity
    4. Key Terms
    5. Section Summary
    6. Key Equations
    7. Chapter Review
      1. Concept Items
      2. Critical Thinking Items
      3. Problems
      4. Performance Task
    8. Test Prep
      1. Multiple Choice
      2. Short Answer
      3. Extended Response
  9. 8 Momentum
    1. Introduction
    2. 8.1 Linear Momentum, Force, and Impulse
    3. 8.2 Conservation of Momentum
    4. 8.3 Elastic and Inelastic Collisions
    5. Key Terms
    6. Section Summary
    7. Key Equations
    8. Chapter Review
      1. Concept Items
      2. Critical Thinking Items
      3. Problems
      4. Performance Task
    9. Test Prep
      1. Multiple Choice
      2. Short Answer
      3. Extended Response
  10. 9 Work, Energy, and Simple Machines
    1. Introduction
    2. 9.1 Work, Power, and the Work–Energy Theorem
    3. 9.2 Mechanical Energy and Conservation of Energy
    4. 9.3 Simple Machines
    5. Key Terms
    6. Section Summary
    7. Key Equations
    8. Chapter Review
      1. Concept Items
      2. Critical Thinking Items
      3. Problems
      4. Performance Task
    9. Test Prep
      1. Multiple Choice
      2. Short Answer
      3. Extended Response
  11. 10 Special Relativity
    1. Introduction
    2. 10.1 Postulates of Special Relativity
    3. 10.2 Consequences of Special Relativity
    4. Key Terms
    5. Section Summary
    6. Key Equations
    7. Chapter Review
      1. Concept Items
      2. Critical Thinking Items
      3. Problems
      4. Performance Task
    8. Test Prep
      1. Multiple Choice
      2. Short Answer
      3. Extended Response
  12. 11 Thermal Energy, Heat, and Work
    1. Introduction
    2. 11.1 Temperature and Thermal Energy
    3. 11.2 Heat, Specific Heat, and Heat Transfer
    4. 11.3 Phase Change and Latent Heat
    5. Key Terms
    6. Section Summary
    7. Key Equations
    8. Chapter Review
      1. Concept Items
      2. Critical Thinking Items
      3. Problems
      4. Performance Task
    9. Test Prep
      1. Multiple Choice
      2. Short Answer
      3. Extended Response
  13. 12 Thermodynamics
    1. Introduction
    2. 12.1 Zeroth Law of Thermodynamics: Thermal Equilibrium
    3. 12.2 First law of Thermodynamics: Thermal Energy and Work
    4. 12.3 Second Law of Thermodynamics: Entropy
    5. 12.4 Applications of Thermodynamics: Heat Engines, Heat Pumps, and Refrigerators
    6. Key Terms
    7. Section Summary
    8. Key Equations
    9. Chapter Review
      1. Concept Items
      2. Critical Thinking Items
      3. Problems
      4. Performance Task
    10. Test Prep
      1. Multiple Choice
      2. Short Answer
      3. Extended Response
  14. 13 Waves and Their Properties
    1. Introduction
    2. 13.1 Types of Waves
    3. 13.2 Wave Properties: Speed, Amplitude, Frequency, and Period
    4. 13.3 Wave Interaction: Superposition and Interference
    5. Key Terms
    6. Section Summary
    7. Key Equations
    8. Chapter Review
      1. Concept Items
      2. Critical Thinking Items
      3. Problems
      4. Performance Task
    9. Test Prep
      1. Multiple Choice
      2. Short Answer
      3. Extended Response
  15. 14 Sound
    1. Introduction
    2. 14.1 Speed of Sound, Frequency, and Wavelength
    3. 14.2 Sound Intensity and Sound Level
    4. 14.3 Doppler Effect and Sonic Booms
    5. 14.4 Sound Interference and Resonance
    6. Key Terms
    7. Section Summary
    8. Key Equations
    9. Chapter Review
      1. Concept Items
      2. Critical Thinking Items
      3. Problems
      4. Performance Task
    10. Test Prep
      1. Multiple Choice
      2. Short Answer
      3. Extended Response
  16. 15 Light
    1. Introduction
    2. 15.1 The Electromagnetic Spectrum
    3. 15.2 The Behavior of Electromagnetic Radiation
    4. Key Terms
    5. Section Summary
    6. Key Equations
    7. Chapter Review
      1. Concept Items
      2. Critical Thinking Items
      3. Problems
      4. Performance Task
    8. Test Prep
      1. Multiple Choice
      2. Short Answer
      3. Extended Response
  17. 16 Mirrors and Lenses
    1. Introduction
    2. 16.1 Reflection
    3. 16.2 Refraction
    4. 16.3 Lenses
    5. Key Terms
    6. Section Summary
    7. Key Equations
    8. Chapter Review
      1. Concept Items
      2. Critical Thinking Items
      3. Problems
      4. Performance Task
    9. Test Prep
      1. Multiple Choice
      2. Short Answer
      3. Extended Response
  18. 17 Diffraction and Interference
    1. Introduction
    2. 17.1 Understanding Diffraction and Interference
    3. 17.2 Applications of Diffraction, Interference, and Coherence
    4. Key Terms
    5. Section Summary
    6. Key Equations
    7. Chapter Review
      1. Concept Items
      2. Critical Thinking Items
      3. Problems
      4. Performance Task
    8. Test Prep
      1. Multiple Choice
      2. Short Answer
      3. Extended Response
  19. 18 Static Electricity
    1. Introduction
    2. 18.1 Electrical Charges, Conservation of Charge, and Transfer of Charge
    3. 18.2 Coulomb's law
    4. 18.3 Electric Field
    5. 18.4 Electric Potential
    6. 18.5 Capacitors and Dielectrics
    7. Key Terms
    8. Section Summary
    9. Key Equations
    10. Chapter Review
      1. Concept Items
      2. Critical Thinking Items
      3. Problems
      4. Performance Task
    11. Test Prep
      1. Multiple Choice
      2. Short Answer
      3. Extended Response
  20. 19 Electrical Circuits
    1. Introduction
    2. 19.1 Ohm's law
    3. 19.2 Series Circuits
    4. 19.3 Parallel Circuits
    5. 19.4 Electric Power
    6. Key Terms
    7. Section Summary
    8. Key Equations
    9. Chapter Review
      1. Concept Items
      2. Critical Thinking Items
      3. Problems
      4. Performance Task
    10. Test Prep
      1. Multiple Choice
      2. Short Answer
      3. Extended Response
  21. 20 Magnetism
    1. Introduction
    2. 20.1 Magnetic Fields, Field Lines, and Force
    3. 20.2 Motors, Generators, and Transformers
    4. 20.3 Electromagnetic Induction
    5. Key Terms
    6. Section Summary
    7. Key Equations
    8. Chapter Review
      1. Concept Items
      2. Critical Thinking Items
      3. Problems
      4. Performance Task
    9. Test Prep
      1. Multiple Choice
      2. Short Answer
      3. Extended Response
  22. 21 The Quantum Nature of Light
    1. Introduction
    2. 21.1 Planck and Quantum Nature of Light
    3. 21.2 Einstein and the Photoelectric Effect
    4. 21.3 The Dual Nature of Light
    5. Key Terms
    6. Section Summary
    7. Key Equations
    8. Chapter Review
      1. Concept Items
      2. Critical Thinking Items
      3. Problems
      4. Performance Task
    9. Test Prep
      1. Multiple Choice
      2. Short Answer
      3. Extended Response
  23. 22 The Atom
    1. Introduction
    2. 22.1 The Structure of the Atom
    3. 22.2 Nuclear Forces and Radioactivity
    4. 22.3 Half Life and Radiometric Dating
    5. 22.4 Nuclear Fission and Fusion
    6. 22.5 Medical Applications of Radioactivity: Diagnostic Imaging and Radiation
    7. Key Terms
    8. Section Summary
    9. Key Equations
    10. Chapter Review
      1. Concept Items
      2. Critical Thinking Items
      3. Performance Task
    11. Test Prep
      1. Multiple Choice
      2. Short Answer
      3. Extended Response
  24. 23 Particle Physics
    1. Introduction
    2. 23.1 The Four Fundamental Forces
    3. 23.2 Quarks
    4. 23.3 The Unification of Forces
    5. Key Terms
    6. Section Summary
    7. Chapter Review
      1. Concept Items
      2. Critical Thinking Items
      3. Performance Task
    8. Test Prep
      1. Multiple Choice
      2. Short Answer
      3. Extended Response
  25. A | Reference Tables
  26. Index

Section Learning Objectives

By the end of this section, you will be able to do the following:

  • Calculate the strength of an electric field
  • Create and interpret drawings of electric fields

Teacher Support

Teacher Support

The learning objectives in this section will help your students master the following standards:

  • (5) The student knows the nature of forces in the physical world. The student is expected to:
    • (C) describe and calculate how the magnitude of the electrical force between two objects depends on their charges and the distance between them.

Section Key Terms

electric field test charge

Teacher Support

Teacher Support

Ask students whether they have seen movies that use the concept of fields as in force fields. Have them describe how such fields work. Describe how gravity can be thought of as a field that surrounds a mass and with which other masses interact. Explain that electric fields are very similar to gravitational fields.

You may have heard of a force field in science fiction movies, where such fields apply forces at particular positions in space to keep a villain trapped or to protect a spaceship from enemy fire. The concept of a field is very useful in physics, although it differs somewhat from what you see in movies.

A field is a way of conceptualizing and mapping the force that surrounds any object and acts on another object at a distance without apparent physical connection. For example, the gravitational field surrounding Earth and all other masses represents the gravitational force that would be experienced if another mass were placed at a given point within the field. Michael Faraday, an English physicist of the nineteenth century, proposed the concept of an electric field. If you know the electric field, then you can easily calculate the force (magnitude and direction) applied to any electric charge that you place in the field.

An electric field is generated by electric charge and tells us the force per unit charge at all locations in space around a charge distribution. The charge distribution could be a single point charge; a distribution of charge over, say, a flat plate; or a more complex distribution of charge. The electric field extends into space around the charge distribution. Now consider placing a test charge in the field. A test charge is a positive electric charge whose charge is so small that it does not significantly disturb the charges that create the electric field. The electric field exerts a force on the test charge in a given direction. The force exerted is proportional to the charge of the test charge. For example, if we double the charge of the test charge, the force exerted on it doubles. Mathematically, saying that electric field is the force per unit charge is written as

E = F q test E = F q test
18.15

where we are considering only electric forces. Note that the electric field is a vector field that points in the same direction as the force on the positive test charge. The units of electric field are N/C.

If the electric field is created by a point charge or a sphere of uniform charge, then the magnitude of the force between this point charge Q and the test charge is given by Coulomb’s law

F= k| Q q test | r 2 F= k| Q q test | r 2

where the absolute value is used, because we only consider the magnitude of the force. The magnitude of the electric field is then

E= F q test = k| Q | r 2 . E= F q test = k| Q | r 2 .
18.16

This equation gives the magnitude of the electric field created by a point charge Q. The distance r in the denominator is the distance from the point charge, Q, or from the center of a spherical charge, to the point of interest.

If the test charge is removed from the electric field, the electric field still exists. To create a three-dimensional map of the electric field, imagine placing the test charge in various locations in the field. At each location, measure the force on the charge, and use the vector equation E = F / q test E = F / q test to calculate the electric field. Draw an arrow at each point where you place the test charge to represent the strength and the direction of the electric field. The length of the arrows should be proportional to the strength of the electric field. If you join together these arrows, you obtain lines. Figure 18.17 shows an image of the three-dimensional electric field created by a positive charge.

This figure shows a sphere with a plus sign at its center and arrows of varying lengths pointing in every direction away from it. Between the arrows to the left of the sphere is the letter E with an arrow above the letter, indicating that E is a vector.
Figure 18.17 Three-dimensional representation of the electric field generated by a positive charge.

Teacher Support

Teacher Support

[BL][OL]Point out that all electric field lines originate from the charge.

[AL]Point out that the number of lines crossing an imaginary sphere surrounding the charge is the same no matter what size sphere you choose. Ask whether students can use this to show that the number of field lines crossing a surface per unit area shows that the electric field strength decreases as the inverse square of the distance.

Just drawing the electric field lines in a plane that slices through the charge gives the two-dimensional electric-field maps shown in Figure 18.18. On the left is the electric field created by a positive charge, and on the right is the electric field created by a negative charge.

Notice that the electric field lines point away from the positive charge and toward the negative charge. Thus, a positive test charge placed in the electric field of the positive charge will be repelled. This is consistent with Coulomb’s law, which says that like charges repel each other. If we place the positive charge in the electric field of the negative charge, the positive charge is attracted to the negative charge. The opposite is true for negative test charges. Thus, the direction of the electric field lines is consistent with what we find by using Coulomb’s law.

The equation E= k| Q |/ r 2 E= k| Q |/ r 2 says that the electric field gets stronger as we approach the charge that generates it. For example, at 2 cm from the charge Q (r = 2 cm), the electric field is four times stronger than at 4 cm from the charge (r = 4 cm). Looking at Figure 18.17 and Figure 18.18 again, we see that the electric field lines become denser as we approach the charge that generates it. In fact, the density of the electric field lines is proportional to the strength of the electric field!

This figure has two parts, each enclosed in a square. At the center of the square on the left is a tiny red circle from which emanate numerous arrows that end on the sides of the square. At the center of the square on the right is a tiny blue circle, and numerous arrows that start from the sides of the square point toward and end at this circle.
Figure 18.18 Electric field lines from two point charges. The red point on the left carries a charge of +1 nC, and the blue point on the right carries a charge of –1 nC. The arrows point in the direction that a positive test charge would move. The field lines are denser as you approach the point charge.

Electric-field maps can be made for several charges or for more complicated charge distributions. The electric field due to multiple charges may be found by adding together the electric field from each individual charge. Because this sum can only be a single number, we know that only a single electric-field line can go through any given point. In other words, electric-field lines cannot cross each other.

Figure 18.19(a) shows a two-dimensional map of the electric field generated by a charge of +q and a nearby charge of −q. The three-dimensional version of this map is obtained by rotating this map about the axis that goes through both charges. A positive test charge placed in this field would experience a force in the direction of the field lines at its location. It would thus be repelled from the positive charge and attracted to the negative charge. Figure 18.19(b) shows the electric field generated by two charges of −q. Note how the field lines tend to repel each other and do not overlap. A positive test charge placed in this field would be attracted to both charges. If you are far from these two charges, where far means much farther than the distance between the charges, the electric field looks like the electric field from a single charge of −2q.

This figure has two parts, each of which displays two small circles and numerous curved arrows. In Part a, the circle on the left is marked “plus q” and the one on the right is marked “minus q”. Numerous curved arrows emanate from the circle on the left, and many of them terminate on the circle on the right. In Part b, both circles are marked “minus q”. Numerous curved arrows coming from outside these circles terminate on them. Between the two circles is a gap, because the arrows terminating on the left circle do not touch those terminating on the right circle.
Figure 18.19 (a) The electric field generated by a positive point charge (left) and a negative point charge of the same magnitude (right). (b) The electric field generated by two equal negative charges.

Teacher Support

Teacher Support

Ask students to interpret the electric field maps. Where is the field strongest? Where is the field weakest? In which direction is the field increasing or decreasing? Where is the field the most uniform? Can they verify that the magnitude of the charges is the same in a given panel? How does the field for the two negative charges differ from that of the positive and negative charges?

Virtual Physics

Probing an Electric Field

This simulation shows you the electric field due to charges that you place on the screen. Start by clicking the top checkbox in the options panel on the right-hand side to show the electric field. Drag charges from the buckets onto the screen, move them around, and observe the electric field that they form. To see more precisely the magnitude and direction of the electric field, drag an electric-field sensor, or E-field sensor from the bottom bucket, and move it around the screen.

Grasp Check

Two positive charges are placed on a screen. Which statement describes the electric field produced by the charges?
  1. It is constant everywhere.
  2. It is zero near each charge.
  3. It is zero halfway between the charges.
  4. It is strongest halfway between the charges.

Watch Physics

Electrostatics (part 2): Interpreting electric field

This video explains how to calculate the electric field of a point charge and how to interpret electric-field maps in general. Note that the lecturer uses d for the distance between particles instead of r. Note that the point charges are infinitesimally small, so all their charges are focused at a point. When larger charged objects are considered, the distance between the objects must be measured between the center of the objects.

Grasp Check

True or false—If a point charge has electric field lines that point into it, the charge must be ositive.

  1. true
  2. false

Worked Example

What is the charge?

Look at the drawing of the electric field in Figure 18.20. What is the relative strength and sign of the three charges?

This diagram shows three small circles and various curved arrows pointing toward or away from them. A circle near the top left corner is labeled “1”, another located east of it is labeled “2”, and a third located roughly south of the first circle is labeled “3”. Some curved arrows emanating from circle 1 terminate on circle 2, and other curved arrows emanating from circle 1 terminate on circle 3.
Figure 18.20 Map of electric field due to three charged particles.

Strategy

We know the electric field extends out from positive charge and terminates on negative charge. We also know that the number of electric field lines that touch a charge is proportional to the charge. Charge 1 has 12 fields coming out of it. Charge 2 has six field lines going into it. Charge 3 has 12 field lines going into it.

Discussion

Although we cannot determine the precise charge on each particle, we can get a lot of information from the electric field regarding the magnitude and sign of the charges and where the force on a test charge would be greatest (or least).

Worked Example

Electric field from doorknob

A doorknob, which can be taken to be a spherical metal conductor, acquires a static electricity charge of q=−1.5nC. q=−1.5nC. What is the electric field 1.0 cm in front of the doorknob? The diameter of the doorknob is 5.0 cm.

Strategy

Because the doorknob is a conductor, the entire charge is distributed on the outside surface of the metal. In addition, because the doorknob is assumed to be perfectly spherical, the charge on the surface is uniformly distributed, so we can treat the doorknob as if all the charge were located at the center of the doorknob. The validity of this simplification will be proved in a later physics course. Now sketch the doorknob, and define your coordinate system. Use +x +x to indicate the outward direction perpendicular to the door, with x=0 x=0 at the center of the doorknob (as shown in the figure below).

This diagram shows the edge of a door in cross-section and a doorknob attached to it. Two axes, perpendicular to each other, originate from the center of the doorknob and point away from it. The horizontal axis is labeled “plus x circumflex” and the vertical axis is labeled “plus y circumflex”. A dot on the horizontal axis beyond the doorknob is labeled with the words “We want to know electric field at this point”.

If the diameter of the doorknob is 5.0 cm, its radius is 2.5 cm. We want to know the electric field 1.0 cm from the surface of the doorknob, which is a distance r=2.5cm+1.0cm=3.5cm r=2.5cm+1.0cm=3.5cm from the center of the doorknob. We can use the equation E= k| Q | r 2 E= k| Q | r 2 to find the magnitude of the electric field. The direction of the electric field is determined by the sign of the charge, which is negative in this case.

Discussion

This seems like an enormous electric field. Luckily, it takes an electric field roughly 100 times stronger ( 3× 10 6 N/C 3× 10 6 N/C ) to cause air to break down and conduct electricity. Also, the weight of an adult is about 70kg×9.8 m/s 2 700N 70kg×9.8 m/s 2 700N, so why don’t you feel a force on the protons in your hand as you reach for the doorknob? The reason is that your hand contains an equal amount of negative charge, which repels the negative charge in the doorknob. A very small force might develop from polarization in your hand, but you would never notice it.

Practice Problems

15.

What is the magnitude of the electric field from 20 cm from a point charge of q = 33 nC?

  1. 7.4 × 103 N/C
  2. 1.48 × 103 N / C
  3. 7.4 × 1012 N / C
  4. 0
16.

A −10 nC charge is at the origin. In which direction does the electric field from the charge point at x + 10 cm ?

  1. The electric field points away from negative charges.
  2. The electric field points toward negative charges.
  3. The electric field points toward positive charges.
  4. The electric field points away from positive charges.

Check Your Understanding

17.

When electric field lines get closer together, what does that tell you about the electric field?

  1. The electric field is inversely proportional to the density of electric field lines.
  2. The electric field is directly proportional to the density of electric field lines.
  3. The electric field is not related to the density of electric field lines.
  4. The electric field is inversely proportional to the square root of density of electric field lines.
18.

If five electric-field lines come out of a +5 nC charge, how many electric-field lines should come out of a +20 nC charge?

  1. five field lines
  2. 10 field lines
  3. 15 field lines
  4. 20 field lines
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