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Physics 2102 QuickTime™ and a decompressor are needed to see this picture. QuickTime™ and a decompressor are needed to see this picture. Jonathan Dowling Physics 2102 Lecture: 03 TUE 26 JAN Electric Fields II QuickTime™ and a decompressor are needed to see this picture. Michael Faraday (1791-1867) What Are We Going to Learn? A Road Map • Electric charge - Electric force on other electric charges - Electric field, and electric potential • Moving electric charges : current • Electronic circuit components: batteries, resistors, capacitors • Electric currents - Magnetic field - Magnetic force on moving charges • Time-varying magnetic field Electric Field • More circuit components: inductors. • Electromagnetic waves - light waves • Geometrical Optics (light rays). • Physical optics (light waves) Coulomb’s Law q1 F12 F21 r12 k | q1 | | q2 | | F12 | 2 r12 q2 For Charges in a Vacuum 2 N m 9 8 . 99 10 k= C2 Often, we write k as: k 1 4 0 with 0 8.85 10 12 2 C 2 Nm E-Field is E-Force Divided by ECharge r r F Definition of E Electric Field: q r k | q1 | | q2 | | F12 | 2 r12 r k | q2 | | E12 | 2 r12 EForce on Charge +q1 P1 Units: F = [N] = [Newton] ; P1 –q2 P2 r E12 E-Field at Point r F12 –q2 P2 E = [N/C] = [Newton/Coulomb] Force on a Charge in Electric Field Definition of Electric Field: Force on Charge Due to Electric Field: r r F E q r r F qE Force on a Charge in Electric Field +++++++++ Positive Charge Force in Same Direction as EE Field (Follows) –––––––––– +++++++++ E –––––––––– Negative Charge Force in Opposite Direction as EField (Opposes) Electric Dipole in a Uniform Field • Net force on dipole = 0; center of mass stays where it is. • Net TORQUE : INTO page. Dipole rotates to line up in direction of E. • | | = 2(qE)(d/2)(sin ) = (qd)(E)sin = |p| E sin = |p x E| • The dipole tends to “align” itself with the field lines. • What happens if the field is NOT UNIFORM?? Distance Between Charges = d Electric Charges and Fields First: Given Electric Charges, We Calculate the Electric Field Using E=kqr/r3. 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=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 Continuous Charge Distribution • Thus Far, We Have Only Dealt With Discrete, Point Charges. q • Imagine Instead That a Charge q Is Smeared Out Over A: q – LINE q – AREA – VOLUME • How to Compute the Electric Field E? Calculus!!! q Charge Density • Useful idea: charge density • Line of charge: charge per unit length = • Sheet of charge: = q/L charge = q/A per unit area = • Volume of charge: per unit volume = charge = 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 = dL dq = dS dq = dV Differential Form of Coulomb’s Law r k q2 E12 2 r12 E-Field at Point r E12 P1 q2 P2 r k dq2 dE12 2 r12 Differential dE-Field at Point r dE12 P1 dq2 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 E r dE s dE P y x dx a q o L dx 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 q o L Line Of Charge: Field on bisector Distance hypotenuse: dE Charge per unit length: P q L 2 1/2 [C/m] k(dq) dE r2 a r r a x 2 dx q k ( dx)a dE y dE cos 2 (a x 2 )3 / 2 x o L a a cos 2 r (a x 2 )1/2 Adjacent Over Hypotenuse Line Of Charge: Field on bisector L/2 L/2 dx x E y k a 2 2 3 / 2 k a 2 2 2 ( a x ) a x a L / 2 L / 2 Integrate: Trig Substitution! Point Charge Limit: L << a 2kL kL kq Ey 2 2 2 2 a a a 4a L 2kL a 4a L 2 2 Line Charge Limit: L >> a 2kL 2k Ey a a 4a2 L2 Units Check! Coulomb’s Law! Nm2 1 C N C m m m Binomial Approximation from Taylor Series: x<<1 1 x n 2 1/ 2 2kL kL L 2 1 2 2 a 2a a 4a L 1 nx 2 kL 1 L kL 2 1 2 ; (L a) a 2 2a a 1/ 2 2 2 2a 2kL 2kL 2k 1 2a 2k 1 1 ; (L a) aL L a 2 L a a 4a 2 L2 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. kdQ dEx dE cos 2 cos R /2 /2 k (Rd ) cos k Ex 0 k Ex R R 2 k Ey R R –Q E 450 x y cosd dQ = Rd d 0 k E 2 R x 2Q/(R) E Ex2 Ey2 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 • 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 y x • Choose symmetric elements • 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/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.