1. Prepared by Dedra Demaree, Georgetown University
The Electric Field
Prepared by
Dedra Demaree, Georgetown University
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2. The Electric Field
Why is it safe to sit in a car during a lightning storm?
How does a lightning rod protect a building?
How does electrocardiography work?
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3. Be sure you know how to:
Find the force that one charged object exerts on another charged object (Section 14.4).
Determine the electric potential energy of a system (Section 14.5).
Explain the differences in the internal structure of electric conductors and dielectrics (Section 14.3).
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4. What's new in this chapter
We learned how to describe electrostatic interactions in two ways: with a force exerted by one charged object on another and with the electric potential energy.
This is only the second interaction we have encountered where forces are exerted without the objects being in direct contact.
How does one charged object "know" about the presence of another when they are not in direct contact?
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5. A model of the mechanism for electrostatic interactions
A model for electric interactions, suggested by Michael Faraday, involves some sort of electric disturbance in the region surrounding a charged object.
Physicists call this electric disturbance an electric field.
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6. Gravitational field due to a single object with mass
We find a mathematical description of the "strength" of Earth's gravitational field at a particular location that is independent of the test mass:
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7. Electric field due to a single point-like charged object
We use a similar approach of test charges to construct a physical quantity for the "strength" of the electric field:
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8. Electric field due to a single point-like charged object
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9. Electric field due to a single point-like charged object
We can interpret this field as follows:
The E field vector at any location points away from the object creating the field if Q is positive, and toward the object creating the field if Q is negative.
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10. Observational experiment
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11. Observational experiment
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12. Superposition principle
When multiple charged objects are present, each object makes its own contribution to the E field.
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13. Using the superposition principle
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14. Conceptual Exercise 15.1
The muscles of the heart continually contract and relax, making the heart an electric dipole with equal-magnitude positive and negative electric charges. Estimate the direction of the E field at position A in the body tissue to the left side of the midpoint between the dipole charges.
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15. E field lines
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16. E field lines
E field lines point away from an area of positive charge and point toward an area of negative charge.
Closer to the charged objects, the lines are closer together; the number of lines per unit area (the density of lines) is larger where the E field is stronger.
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17. Tip
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18. E field lines
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19. Tip
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20. Conceptual Exercise 15.2
Draw E field lines for a large, uniformly charged plate of glass.
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21. Determining the E field produced by given source charges
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22. Example 15.3
Two small metal spheres attached to insulating stands reside on a table a distance d apart. The left sphere has positive charge +q and the right sphere has negative charge −q. Determine the magnitude and direction for the E field at a distance d above the center of the line connecting the spheres.
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23. Problem-solving strategy: Incorporating the E field into Newton's second law
In the "Simplify and diagram" step, be sure to determine the E field produced by the environment. Is it produced by point-like charges (making it nonuniform) or by large charged plates (making it uniform)?
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24. Example 15.5
Inside an inkjet printer, a tiny ball of black ink of mass 1.1 x 10−11 kg with charge −6.7 x 10−12 C moves horizontally at a speed of 40 m/s. The ink ball enters an upward-pointing uniform E field of magnitude 1.0 x 104 N/C produced by a negatively charged plate above and a positively charged plate below. The plates deflect the ink ball so that it lands at a particular spot on a piece of paper. Determine the deflection of the ink ball after it travels m in the E field.
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25. The V field
Can we describe electric fields using the concepts of work and energy?
To do so, we need to describe the electric field not as a force-related E field, but as an energy-related field.
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26. Electric potential due to a single charged object
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27. Tip
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28. The superposition principle and the V field due to multiple charges
where Q1, Q2, Q3, … are the source charges (including their signs) creating the field and r1, r2, r3, … are the distances between the source charges and the location where we are determining the V field.
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29. Quantitative Exercise 15.6
Suppose that the heart's dipole charges −Q and +Q are separated by distance d. Write an expression for the V field due to both charges at point A, a distance d to the right of the +Q charge.
Simplify and diagram.
Represent mathematically.
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30. Finding the electric potential energy when the V field is known
If we know the electric potential at a specific location, we can rearrange the definition of the V field to determine the electric potential energy:
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31. Potential difference
The value of the electric potential depends on the choice of zero level, so we often use the difference in electric potential between two points.
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32. Particles in a potential difference
A positively charged object accelerates from regions of higher electric potential toward regions of lower potential (like an object falling to lower elevation in Earth's gravitational field).
A negatively charged particle tends to do the opposite, accelerating from regions of lower potential toward regions of higher potential.
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33. Example 15.7
Inside an X-ray machine is a wire (called a filament) that, when hot, ejects electrons. Imagine one of those electrons, now located outside the wire. It starts at rest and accelerates through a region where the V field increases by 40,000 V. The electron stops abruptly when it hits a piece of tungsten at the other side of the region, producing X-rays. How fast is the electron moving just before it reaches the tungsten?
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34. Equipotential surfaces: Representing the V field
The lines represent surfaces of constant electric potential V, called equipotential surfaces.
The surfaces are spheres (they look like circles on a two-dimensional page).
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35. Equipotential surfaces and E field
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36. Contour maps: An analogy for equipotential surfaces
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37. Deriving a relation between the E field and ΔV
We attach a small object with charge +q to the end of a very thin wooden stick and place the charged object and stick in the electric field produced by the plate.
The only energy change is the system's electric potential energy, because the positively charged object moves farther away from the positively charged plate.
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38. Deriving a relation between the E field and ΔV
Applying the generalized work-energy equation, we get:
Equivalently, the component of the E field along the line connecting two points on the x-axis is the negative change of the V field divided by the distance between those two points:
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39. Conceptual Exercise 15.8
Can you think of locations relative to charge distributions where:
The V field at a particular location is zero but the E field is not?
The E field is zero but the V field is not zero?
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40. Testing the relation between the E field and ΔV
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41. Electric field of a charged conductor
Free electrons in a conductor are quickly redistributed until equilibrium is reached, at which point the E field inside the conductor and parallel to its surface becomes zero.
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42. Electric field outside a charged conductor
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43. Grounding
Grounding discharges an object made of conducting material by connecting it to Earth.
Electrons will move between and within the spheres until the V field on the surfaces of and within both spheres achieves the same value.
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44. Uncharged conductor in an electric field: Shielding
The free electrons inside the object become redistributed due to electric forces, until the E field within the conducting object is reduced to zero.
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45. Uncharged conductor in an electric field: Shielding
The interior is protected from the external field—an effect called shielding.
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46. Dielectric materials in an electric field
If an atom in a dielectric material resides in a region with an external electric field, the nucleus and the electrons are displaced slightly in opposite directions until the force that the field exerts on each of them is balanced by the force they exert on each other.
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47. Polar water molecules in an external electric field
Some molecules, such as water, are natural electric dipoles even when the external E field is zero.
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48. E field inside a dielectric
A dielectric material cannot completely shield its interior from an external electric field, but it does decrease the field.
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49. E field inside a dielectric
Physicists use a physical quantity to characterize the ability of dialectrics to decrease the E field:
The dielectric constant κ
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50. Dielectric constants for different types of materials
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51. Electric force and dielectrics
The force that object 1 exerts on object 2 is reduced by κ compared with the force it would exert in a vacuum.
Inside the dielectric material, Coulomb's law is now written as:
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52. Salt dissolves in blood but not in air
When salt is placed in water or blood:
Many more collisions occur between molecules than between molecules and air; these can break an ion free from the crystal.
Any ions that become separated do not exert nearly as strong as an attractive force on each other because of the dielectric effect.
The random kinetic energy of the liquid is sufficient to keep the sodium and chlorine ions from recombining, allowing the nervous system to use the freed sodium ions to transmit information.
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53. Tip
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54. Capacitors
A capacitor consists of two conducting surfaces separated by a nonconducting material.
The role of a capacitor is to store electric potential energy.
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55. Capacitors (Cont'd)
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56. Capacitors
If we consider the capacitor plates to be large flat conductors, charge should be distributed evenly on the plates.
The magnitude of the E field between the plates relates to the potential difference from one plate to the other and the distance separating them
To double the E field, the charge on other plates has to double.
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57. Capacitors
The proportionality constant C in this equation is called the capacitance of the capacitor.
The unit of capacitance is 1 coulomb/volt = 1 farad (in honor of Michael Faraday).
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58. Capacitance of a capacitor
A capacitor with larger-surface-area plates should be able to maintain more charge separation because there is more room for the charge to spread out.
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59. Capacitance of a capacitor
A larger distance between the plates leads to a smaller-magnitude E field between the plates. Because the magnitude of this E field is proportional to the amount of electric charge on the plates, a larger plate separation leads to a smaller-magnitude electric charge on the plates.
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60. Capacitance of a capacitor
Material between the plates with a large dielectric constant becomes polarized by the electric field between the plates. Thus more charge moves onto capacitor plates that are separated by material of high dielectric constant.
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61. Capacitance of a capacitor
The capacitance of a particular capacitor should increase if the surface area A of the plates increases, decrease if the distance d between them is increased, and increase if the dielectric constant k of the material between them increases:
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62. Tip
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63. Quantitative Exercise 15.9
Estimate the capacitance of your physics textbook, assuming that the front and back covers (area A = m2, separation d = m) are made of a conducting material. The dielectric constant of paper is approximately 6.0.
Determine what the potential difference must be across the covers for the textbook to have a charge separation of 10−6 C (one plate has charge +10−6 C and the other has charge −10−6 C).
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64. Body cells as capacitors
Cells, including nerve cells, have capacitor-like properties.
The conducting "plates" are the fluids on either side of a moderately nonconducting cell membrane.
In this membrane, chemical processes cause ions to be "pumped" across the membrane.
As a result, the membrane's inner surface becomes slightly negatively charged, while the outer surface becomes slightly positively charged.
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65. Example 15.10 Estimate: The capacitance C of a single cell.
The charge separation q of all of the membranes of the human body's 1013 cells.
Assume that each cell has a surface area of A = 1.8 x 10−9 m2, a membrane thickness of d = 8.0 x 10−9 m, ΔV = across the membrane wall, and a membrane dielectric constant κ = 8.0.
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66. Energy of a charged capacitor
To determine the electric potential energy in a charged capacitor, we start with an uncharged capacitor and then calculate the amount of work that must be done on the capacitor to move electrons from one plate to the other.
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67. Energy of a charged capacitor
The process of charging a capacitor is similar to stretching a spring: at the beginning, a smaller force is needed to stretch the spring by a certain amount compared to the much greater force needed when the spring is already stretched.
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68. Quantitative Exercise 15.11
In Example 15.10, we estimated that the total charge separated across all the cell membranes in a human body was about 11 C. Recall that the potential difference across the cell membranes is V. Estimate the work that must be done to separate the charges across the membranes of the body's approximately 1013 cells.
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69. Energy density of electric field
To have a measure of energy independent of the capacitor volume, we will use the physical quantity of energy density.
This energy density quantifies the electric potential energy stored in the electric field per cubic meter of volume.
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70. Tip
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71. Tip
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72. Electrocardiography
An electric charge separation occurs when muscle cells in the heart contract during the pumping process.
As each muscle cell contracts, positive and negative charges separate.
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73. Conceptual Exercise 15.12
The first figure shows a simplified electric dipole charge distribution on a heart at one instant during a heartbeat and two ECG pads on opposite shoulders of the person's body. What will these pads measure at that particular instant?
Draw E field vectors produced by the heart's dipole charge, representing the electric field at the location of the dot in the figure.
Determine the direction of the forces exerted by the electric field on a positive sodium ion and on a negative chlorine ion in the body tissue at that location.
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74. Lightning
When the E field in air or in some other material is very large, free electrons accelerate and quickly acquire enough kinetic energy to ionize atoms and molecules in their path when colliding with them.
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75. Lightning rods
Dielectric breakdown occurs between the cloud and the lightning rod.
Drawing lightning to the rod and away from the building prevents damage to the building and its inhabitants.
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76. Summary
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77. Summary
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78. Summary
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79. Summary
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