1. Physics 122B Electricity and Magnetism
Lecture 3 (Knight: ) Electric Field, Field Lines and Charge Distributions
March 30, 2007
Martin Savage

2. Electric Charge Distributions
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Physics 122B - Lecture 3

3. Electric Field of a Point Charge
Positive
Negative
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4. Electric Field Superpositionq
Therefore, the electric field of any charge distribution can be calculated by dividing the distribution into point charges and superimposing these.
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Physics 122B - Lecture 3

5. The Electric Field of Multiple Point Charges
Component electric fields may be superimposed to obtain the net field. This is very useful in calculating the field produced by a number of point charges.
This is really three equations:
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Physics 122B - Lecture 3

6. The E-Field of 3 Point Charges in the plane of reflection symmetry
q1=q2=q3=q
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7. The E-Field of 3 Point Charges in the plane of reflection symmetry
Near-field limit:
Far-field limit:
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8. The Electric Dipole Permanent Dipole Induced Dipole 11/13/2018
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9. Electric Field of a Dipole
Y >> s
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10. Field Map vs. Field LinesQ
Field lines start on positive charges.
weak -Q
Field lines stop on negative charges. 2Q
More charge Þ more field lines.
strong
Field lines never cross.
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Physics 122B - Lecture 3

11. A Prelude to Maxwell’s Equations
Suppose you come across a vector field that looks something like this.
What are the identifiable structures in this field?
1. An “outflow” structure:
2. An “inflow” structure:
3. An “clockwise circulation” structure:
4. An “counterclockwise
circulation” structure:
Maxwell’s Equations will tell us that the “flow” structures are charges (+ and -) and the “circulation” structures …. we will come back to this later…do no occur in static situations
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12. The Dipole Field Field Map Field Lines 11/13/2018
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13. Two Positive Charges Field Map Field Lines 11/13/2018
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14. Charge Distributions
Symmetry Principle: An electric field must have the same symmetries as the charge distribution that produced it.
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15. Types of Symmetry A (yes) A E (no) E
Full rotational symmetry: any rotation about any axis does not change object: Sphere
Translational symmetry: translation along a line does not change object.
Cylindrical symmetry: any rotation about one axis does not change object: Cylinder
Reflectional symmetry: mirror image reflection is same as object.
A (yes) A
E (no) E
720
Partial rotational symmetry: 720 rotation about one axis does not change object: Star
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Physics 122B - Lecture 3

16. Computing E-Fields of Charged Objects Using Coulomb’s Law
Choose a coordinate system that will facilitate integration.
Use any applicable symmetries to set E-field components to zero or equal to each other.
Break up the object into point-like elements.
Write the Coulomb’s Law contribution to the E-field from a representative point-like element.
Integrate over the entire object to get the E-field.
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Physics 122B - Lecture 3

17. Example: E-Field of a Charged Line
A thin uniformly charged rod of length L has a total charge Q. Find the electric field at a distance r from the axis of the rod in the plane that bisects the rod.
Strategy: Break up the rod into small charge elements, use Coulomb’s Law to calculate the E field of each, and add them up.
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Physics 122B - Lecture 3

18. Example: E-Field of a Charged Line (2)
Consider the x component of the electric field contribution dEx produced by a small segment i of the rod located at yi with length dy. Its charge DQi will be (Q/L)dy.
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Physics 122B - Lecture 3

19. An Infinite Line of Charge…the limit of a finite line-charge
Top view; field lines spread in a plane +
Side view; field lines are parallel.
Field of an infinite line of charge falls off as 1/r (not 1/r2).
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Physics 122B - Lecture 3

20. Question
Which of the following actions will increase the electric field strength at the position of the dot?
(a) Rod longer, net charge the same; (b) Rod shorter, net charge the same; (c) Rod wider, net charge the same; (d) Rod narrower, net charge the same; (e) Move dot farther from rod.
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Physics 122B - Lecture 3

21. Computing Charged Object E-Fields Using Coulomb’s Law
Choose a coordinate system that will facilitate integration.
Use any applicable symmetries to set E-field components to zero or equal to each other.
Break up the object into point-like elements.
Write the Coulomb’s Law contribution to the E-field from a representative point-like element.
Integrate over the entire object to get the E-field.
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Physics 122B - Lecture 3

22. The E-Field of a Charged Ring
A thin uniformly charged ring of radius R has a total charge Q. Find the electric field on the axis of the ring .. the z-axis …(perpendicular to the page).
The linear charge density of the ring is l = Q/(2pR). The system has cylindrical symmetry for rotations about the axis, so along the z-axis Ex=Ey=0 and we need only to find Ez. Consider a small segment of the circumference of the ring of width dl = R df. The contribution to the electric field on the z-axis is :
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Physics 122B - Lecture 3

23. The on-axis E-Field of a Charged Ring (2)
Near-field limit: linear
Far-field limit: 1/r2 (same as point charge)
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24. End of Lecture 3
Before the next lecture, read Knight, Chapters 26.4 through 26.5
Lecture Homework #1 is on the Tycho system and is due at 10pm on Wednesday