1. Physics 2113 Lecture 07: FRI 30 JAN
Jonathan Dowling
Physics Lecture 07: FRI 30 JAN
Electric Fields II
Charles-Augustin
de Coulomb
( )

2. Direction of Electric Field Lines
E-Field Vectors Point Away from Positive Charge — Field Source! Positives PUSH!
E-Field Vectors Point Towards Negative Charge — Field Sink! Negatives Pull!

3. Electric Field of a Dipole
Electric dipole: two point charges +q and –q separated by a distance d
Common arrangement in Nature: molecules, antennae, …
Note axial or cylindrical symmetry
Define “dipole moment” vector p: from –q to +q, with magnitude qd
Cancer, Cisplatin and electric dipoles:

4. Electric Field On Axis of Dipole-q +q

5. Electric Field On Axis of Dipole
p = qa
“dipole moment”
a VECTOR
What if x>> a? (i.e. very far away)
E = p/r3 is actually true for ANY point far from a dipole (not just on axis)

6. Continuous Charge Distribution
Thus Far, We Have Only Dealt With Discrete, Point Charges.
Imagine Instead That a Charge q Is Smeared Out Over A:
LINE
AREA
VOLUME
How to Compute the Electric Field E? Calculus!!! q q q q

7. Charge Density λ = q/L Useful idea: charge density
Line of charge: charge per unit length = λ
Sheet of charge: charge per unit area = σ
Volume of charge: charge per unit volume = ρ
σ = q/A
ρ = q/V

8. Computing Electric Field of Continuous Charge Distributiondq
Approach: Divide the Continuous Charge Distribution Into Infinitesimally Small Differential Elements
Treat Each Element As a POINT Charge & Compute Its Electric Field
Sum (Integrate) Over All Elements
Always Look for Symmetry to Simplify Calculation!
dq = λ dL
dq = σ dS
dq = ρ dV

9. Differential Form of Coulomb’s Law
E-Field at Point q2 P1 P2
Differential dE-Field at Point
dq2 P1

10. Field on Bisector of Charged Rod
Uniform line of charge +q spread over length L
ICPP: What is the direction of the electric field at a point P on the perpendicular bisector?
(a) Field is 0.
(b) Along +y
(c) Along +x dx dx P y x a
Choose symmetrically located elements of length dq = λdx
x components of E cancel q o L

11. Line of Charge: Quantitative
Uniform line of charge, length L, total charge q
Compute explicitly the magnitude of E at point P on perpendicular bisector
Showed earlier that the net field at P is in the y direction — let’s now compute this! P y x a q o L

12. Line Of Charge: Field on bisector
Distance hypotenuse: dE
Charge per unit length:
[C/m] P a r dx q o x
Adjacent
Over
Hypotenuse L

13. Line Of Charge: Field on bisector
Integrate: Trig Substitution!
Point Charge Limit: L << a
Line Charge Limit: L >> a
Units Check!
Coulomb’s
Law!

14. Binomial Approximation from Taylor Series:

15. ICPP: What is the direction of the field at point P?dx dx P
ICPP: What is the direction of the field at point P?
Along +x
Along –x
Along +y
Along -y y x R ✔ q o L

17. ICPP: Question What is the direction of the electric field at point P?
Positives PUSH / Source of Field
Negatives PULL / Sink of Field

18. Problem Calculate the magnitude of the electric field at point P. dE xdx dE