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 q 2 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 nits F N Newton EForce on Charg e E Field q1 P1 P1 q2 P2 r E12 at Point r F12 q2 P2 E N C Newton Coulom Force on a Charge in Electric Field r Definition of Electric Field Force on Charge Due to Electric Field 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 E 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 Charge Produces E Field We Calculate the Electric Field Using E kqr r3 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 Continuous Charge Distribution Thus Far We Have Only Dealt With Discrete Point Charges Imagine Instead That a Charge q q Is Smeared Out Over A q LINE AREA q 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 Sheet of charge q A 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 r k q2 E12 2 r12 E Field at Point r E12 P1 q2 P2 r k dq2 dE12 2 r12 Differentia l 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 E r dE s 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 r 2 a x P q Charge per unit length L r k dq dE 2 r dE a dx x o L q 2 1 2 C m k dx a dE y dE cos 2 a x 2 3 2 a a cos 2 r a x2 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 2k L a 4a 2 L2 Point Charge Limit L a Line Charge Limit L a k L kq Ey 2 2 2 2 a a a 4a L 2k L 2k L 2k Ey a a 4a 2 L2 Units Check Coulomb s Law Nm2 1 C N C m m m Binomial Approximation from Taylor Series x 1 n 1 x 1 nx 2 1 2 2k L k L L 2 1 2 2 a 2a a 4a L 2 k L 1 L k L 2 1 2 L a a 2 2a a 2 1 2 2 2a 2k L 2k L 2k 1 2a 2k L a 1 1 2 2 aL L a 2 L a a 4a L 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 dE x dE cos 2 cos R 2 k Rd cos k Ex 2 R R 0 k Ex R k Ey R Q E 450 x y dQ Rd d 2 cos d 0 k 2Q E 2 E E E R R 2 x x 2 y 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 element 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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