Physics 122B Electricity and Magnetism Lecture 3 (Knight: 26.2-3) Electric Field, Field Lines and Charge Distributions March 30, 2007 Martin Savage

Electric Charge Distributions 11/13/2018 Physics 122B - Lecture 3

Electric Field of a Point Charge Positive Negative 11/13/2018 Physics 122B - Lecture 3

Electric Field Superposition q Therefore, the electric field of any charge distribution can be calculated by dividing the distribution into point charges and superimposing these. 11/13/2018 Physics 122B - Lecture 3

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: 11/13/2018 Physics 122B - Lecture 3

The E-Field of 3 Point Charges in the plane of reflection symmetry q1=q2=q3=q 11/13/2018 Physics 122B - Lecture 3

The E-Field of 3 Point Charges in the plane of reflection symmetry Near-field limit: Far-field limit: 11/13/2018 Physics 122B - Lecture 3

The Electric Dipole Permanent Dipole Induced Dipole 11/13/2018 Physics 122B - Lecture 3

Electric Field of a Dipole Y >> s 11/13/2018 Physics 122B - Lecture 3

Field Map vs. Field Lines Q 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. 11/13/2018 Physics 122B - Lecture 3

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 11/13/2018 Physics 122B - Lecture 3

The Dipole Field Field Map Field Lines 11/13/2018 Physics 122B - Lecture 3

Two Positive Charges Field Map Field Lines 11/13/2018 Physics 122B - Lecture 3

Charge Distributions Symmetry Principle: An electric field must have the same symmetries as the charge distribution that produced it. 11/13/2018 Physics 122B - Lecture 3

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 11/13/2018 Physics 122B - Lecture 3

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. 11/13/2018 Physics 122B - Lecture 3

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. 11/13/2018 Physics 122B - Lecture 3

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. 11/13/2018 Physics 122B - Lecture 3

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). 11/13/2018 Physics 122B - Lecture 3

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. 11/13/2018 Physics 122B - Lecture 3

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. 11/13/2018 Physics 122B - Lecture 3

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 : 11/13/2018 Physics 122B - Lecture 3

The on-axis E-Field of a Charged Ring (2) Near-field limit: linear Far-field limit: 1/r2 (same as point charge) 11/13/2018 Physics 122B - Lecture 3

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