Physics 2102 Jonathan Dowling Physics 2102 Lecture 04

Physics 2102 Jonathan Dowling Physics 2102 Lecture: 04 WED 21 JAN Electric Fields II 11/28/2020 Version: 11/28/2020 Michael Faraday (1791 -1867)

Electric Charges and Fields First: Given Electric Charges, We Calculate the Electric Field Using E=kqr/r 3. Charge Produces EField Example: the Electric Field Produced By a Single Charge, or by a Dipole: Second: Given an Electric Field, We Calculate the Forces on Other Charges Using F=q. E 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

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

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

Computing Electric Field of Continuous Charge Distribution • Approach: Divide the Continuous Charge Distribution Into Infinitesimally Small Differential Elements dq • Treat Each Element As a POINT Charge & Compute Its Electric Field • Sum (Integrate) Over All Elements • Always Look for Symmetry to Simplify Calculation! dq = d. L dq = d. S dq = d. V

Differential Form of Coulomb’s Law E-Field at Point q 2 P 1 Differential d. E-Field at Point P 1 P 2 dq 2

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 • Choose symmetrically located elements of length dq = dx • x components of E cancel P y x dx a o L dx q

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 a x o L q

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

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

Binomial Approximation from Taylor Series: x<<1

Example — Arc of Charge: Quantitative y • 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. –Q E 450 x y d. Q = Rdq dq q x = 2 Q/(p. R)

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 z y x

Example : Arc of Charge y • 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? x (a) Field is 0. (b) Along +y • Choose symmetric elements (c) Along -y • x components cancel

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/r 2 • Electric field of a dipole: E~kp/r 3 • 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.