Physics 2102 Jonathan Dowling Physics 2102 Lecture 04 WED 21 JAN Electric Fields II January 19 09 Version 1 19 09 Michael Faraday 1791 1867 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 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 dL dq dS dq dV Differential Form of Coulomb s Law k q2 E12 2 r12 E Field at Point E12 P1 q2 P2 k dq2 dE12 2 r12 Differential dE Field at Point 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 dE dE 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 dE P Charge per unit length dx x o L q L 2 1 2 k dq dE 2 d a d d a x 2 q k dx a dE y dE cos 2 a x 2 3 2 a cos 2 2 1 2 a x 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 2k L k L kq Ey 2 2 2 2 a a a 4a L Coulomb s Law 2k L 2 2 a 4a L Line Charge Limit L a 2k L 2k Ey a a 4a 2 L2 Units Check Nm2 1 C N C m m m Binomial Approximation from Taylor Series x 1 1 x n 2 1 2 k L L 2k L 2 1 2 2 a 2a a 4a L 1 nx 2 k L 1 L k L 0 2 1 0 2 L a a 2 2a a 2 1 2 2 2a 2k L 2k 1 2a 2k 2k L L a 1 0 1 0 2 2 aL L a 2 L a a 4a L Example Arc of Charge Quantitative 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 dE x dE cos 2 cos R 2 Ex 0 k Rd cos k 2 R R k Ex R k Ey R y Q E 450 x y dQ Rd d 2 cos d 0 k E 2 R x 2Q 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 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
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