Physics 2102 Lecture: 04 WED 21 JAN

1. Physics 2102 Jonathan Dowling Physics 2102 Lecture: 04 WED 21 JAN Electric Fields II Michael Faraday (1791-1867) Version: 11/14/2014

2. First: Given Electric Charges, We Calculate the Electric Field Using E=kqr/r3. Example: the Electric Field Produced By a Single Charge, or by a Dipole: Charge Produces E-Field Second: Given an Electric Field, We Calculate the Forces on Other Charges Using F=qE Examples: Forces on a Single Charge When Immersed in the Field of a Dipole, Torque on a Dipole When Immersed in an Uniform Electric Field. E-Field Then Produces Force on Another Charge Electric Charges and Fields

3. 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

4.  = q/L  = q/A  = q/V Charge Density • Useful idea: charge density • Line of charge: charge per unit length =  • Sheet of charge: charge per unit area =  • Volume of charge: charge per unit volume = 

5. Computing Electric Field of Continuous Charge Distribution dq • 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 =  dL dq =  dS dq =  dV

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

7. dx dx Field on Bisector of Charged Rod • Uniform line of charge +q spread over length L • 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 P y x a • Choose symmetrically located elements of length dq = dx • x components of E cancel q o L

8. y x Line of Charge: Quantitative P • 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! a q o L

9. dE Line Of Charge: Field on bisector Distance hypotenuse: Charge per unit length: P a d dx q o x Adjacent Over Hypotenuse L

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

11. Binomial Approximation from Taylor Series: x<<1

12. y Example — Arc of Charge: Quantitative –Q E • Figure shows a uniformly charged rod of charge -Q bent into a circular arc of radius R, centered at (0,0). • Compute the direction & magnitude of E at the origin. 450 x y dQ = lRdq dq q x l = 2Q/(pR)

13. z y x Example : Field on Axis of Charged Disk • A uniformly charged circular disk (with positive charge) • What is the direction of E at point P on the axis? (a) Field is 0 (b) Along +z (c) Somewhere in the x-y plane P

14. y x • Choose symmetric elements • x components cancel Example : Arc of Charge • Figure shows a uniformly charged rod of charge –Q bent into a circular arc of radius R, centered at (0,0). • What is the direction of the electric field at the origin? (a) Field is 0. (b) Along +y (c) Along -y

15. Summary • The electric field produced by a system of charges at any point in space is the force per unit charge they produce at that point. • We can draw field lines to visualize the electric field produced by electric charges. • Electric field of a point charge: E=kq/r2 • Electric field of a dipole: E~kp/r3 • An electric dipole in an electric field rotates to align itself with the field. • Use CALCULUS to find E-field from a continuous charge distribution.