Electrostatics: Point Charges

Transcription

1 9/1/018 Electrostatics: Point Charges EE1 Electromagnetic Field Theory Outline Conditions for Electrostatics Maxwell s Equations for Electrostatics Visualizing the Electric Flux Around Point Charges uantifying the Electric Flux Around Point Charges Multiple Point Charges Force on a Charge Electrostatics -- Point Charges Slide 1

2 9/1/018 Maxwell s Equations for Electrostatics Maxwell s Equations at DC Maxwell s Equations B E t D H J t D v B 0 Constitutive Relations D E B H Electrostatic Equations E 0 D v D E Magnetostatic Equations H J B 0 B H Electrostatics -- Point Charges Slide 4

3 9/1/018 Maxwell s Equations for Electrostatics Faraday s Law E 0 Gauss Law The divergence of D is equal to charge density v. D v Constitutive Relation D E D is proportional to E scaled by []. The curl of E is zero. The electric field is irrotational and forms essentially straight lines. We will only have electric fields if we have charges. Electric fields start and end on charges. Electric fields do not see permeability. Electrostatics -- Point Charges Slide 5 Visualizing the Electric Flux Around Point Charges

4 9/1/018 Visualizing the Electric Flux Around Point Charges We start with a positive charge. Electrostatics -- Point Charges Slide 7 Visualizing the Electric Flux Around Point Charges The electric field around the charge is a smooth and continuous phenomenon, like a cloud or fog. Electrostatics -- Point Charges Slide 8 4

5 9/1/018 Visualizing the Electric Flux Around Point Charges The electric field is also a vector and has a direction associated with every point. Only a few discrete points are visualized here. Electrostatics -- Point Charges Slide 9 Visualizing the Electric Flux Around Point Charges If we trace the direction of the electric flux, continuous lines can be drawn. These are called electric field lines. Electrostatics -- Point Charges Slide 10 5

6 9/1/018 Visualizing the Electric Flux Around Point Charges Typically, the field around a charge is visualized using only the electric field lines. CAUTION: The concept of field lines might imply to some people that the electric field only exists on the lines. This is completely not true! The lines do not exist. They are simply a mathematical tool to help us keep track of the direction of the fields. Electrostatics -- Point Charges Slide 11 Visualizing the Electric Flux Around Point Charges Typical way electric flux is visualized. Less misleading way to visualize electric flux? Electrostatics -- Point Charges Slide 1 6

7 9/1/018 Electric Flux in D Electrostatics -- Point Charges Slide 1 Direction of Field The electric flux points away from positive charge and toward negative charge. Electrostatics -- Point Charges Slide 14 7

8 9/1/018 Direction of Field When both types of charges are present, the flux bends or fringes. Electrostatics -- Point Charges Slide 15 Direction of Field The fringing changes shape depending on the size of the charges. Electrostatics -- Point Charges Slide 16 8

9 9/1/018 uantifying the Electric Flux Around Point Charges uantifying the Electric Field Around Charges The electric flux density D is the quantity most closely associated with electric charge. D 4 R a ˆ R D electric flux density C m total charge of point C R distance from charge m aˆ unit vector in radial direction no units R Observation point R aˆr D Electrostatics -- Point Charges Slide 18 9

10 9/1/018 Derivation of Field Around a Charge Apply Gauss law to a point charged located at the origin. Dds S Choose spherical coordinates and pick the observation point to lie at (R,0.0). Due to symmetry, D will only have an r component. D ˆ sin ˆ rar R ddar DR r d sind d DR cos cos0 DR DR r sin 0 Solve for D. 4 Dr R r r D D a 4 R 4 ˆ r R 4 r Electrostatics -- Point Charges Slide 19 1/R Dependence The amplitude of the electric field decreases with distance from the charge. D ar ˆ 4 R 1 R Amplitude of D decays as 1/R. Electrostatics -- Point Charges Slide 0 10

11 9/1/018 Direction of Field The electric flex points away from positive charge and toward negative charge. Electrostatics -- Point Charges Slide 1 More Useful Equation for ˆ r r D a R 4 R 4 r r D electric flux density C m total charge of point C R distance from charge m r observation point m r position of charge m aˆ unit vector in radial direction no units R CAUTION: It is easy to forget 1/R dependence when using this form of the equation, or to think it is 1/ R. D Observation point D Position of charge Electrostatics -- Point Charges Slide aˆ R r r r r r r 11

12 9/1/018 Electric Field Intensity The electric field intensity E and electric flux density D differ by only a constant. D E ˆ r r E a R 4 R 4 r r CAUTION! Many textbooks give this equation as E 4 R 0 aˆ R This is confusing and has a built in assumption that the field is being observed in free space. E Observation point E Position of charge Electrostatics -- Point Charges Slide r r Example # Calculate the electric field intensity at. r y at r.0 C r 1ˆ ˆ ax ay r 6aˆ 4aˆ x air y r x E Solution First, we write the equation for the electric flux density at r due to the charge. r r D 4 r r Second, it is usually convenient to solve the vector part of this equation first. r r 6ˆ 4ˆ 1ˆ ˆ 5ˆ ˆ ax ay ax ay ax ay r r ˆ ˆ 5ax ay aˆ ˆ x ay r r 5 m Third, the electric flux density is C ˆ ˆ 1 D.010 ax ay m 4 nc 50.aˆ 0.1 aˆ x y m Last, the electric field intensity is nc D 50.aˆ 0.1 ˆ x ay m E 5.7aˆ. ˆ 1 F x a r m Electrostatics -- Point Charges Slide 4 y kv m 1

13 9/1/018 Multiple Point Charges Single Charge Electrostatics -- Point Charges Slide 6 1

14 9/1/018 Two Charges Electrostatics -- Point Charges Slide 7 Three Charges Electrostatics -- Point Charges Slide 8 14

15 9/1/018 Craziness Electrostatics -- Point Charges Slide 9 Multiple Charges in D Electrostatics -- Point Charges Slide 0 15

16 9/1/018 Total Electric Flux Density D Superposition the total electric flux arising from multiple point charges is the sum of the electric flux produced by the charges independently. D D a r r N N N i i i ˆ Total i Ri i 1 i 1 4 R i i1 4 r r N total number of point charges DTotal ETotal Electrostatics -- Point Charges Slide 1 i Example # Given the following three point charges, calculate the electric flux density at the origin. 1 0 C at r 1.0aˆ 1.5aˆ 1.0aˆ 1 ˆ ˆ ˆ 100 C at r.0a 1.0a 1.5a 75 C at r 1.0aˆ 1.5aˆ 1.0aˆ D 0,0,0? x y z x y z x y z Electrostatics -- Point Charges Slide 16

17 9/1/018 Example # Step 1 Sketch the problem Step Organize the problem The total electric flux density is the sum of the electric flux density due to each charge separately. D D D D 1 Observation point (0,0,0) Electrostatics -- Point Charges Slide Example # Step Calculate D 1 1 r r 1 D1 4 r r 1 1 r r r r r 0, 0, 0 1, 1.5,1 1,1.5, 1 1 / r r r ,1.5, ,0.17, 0.11 r r 0 D 0.11,0.17, ,0.41, 0.7 C m 4 4 Electrostatics -- Point Charges Slide 4 17

18 9/1/018 Example # Step 4 Calculate D r r D 4 r r r r 0, 0, 0,1, 1.5, 1,1.5 r r, 1,1.5 / r r , 0.051, r r 100 D 0.10, 0.051, ,0.41, 0.61 C m 4 r r 4 Electrostatics -- Point Charges Slide 5 Example # Step 5 Calculate D r r D 4 r r r r 0, 0, 0 1, 1.5, 1 1,1.5,1 r r 1,1.5,1 / r r , 0.17, 0.11 r r 75 D 0.11,0.17, ,1.0,0.68 C m 4 r r 4 Electrostatics -- Point Charges Slide 6 18

19 9/1/018 Example # Step 6 Calculate the total electric flux density D D D D 1 0.7,0.41, ,0.41, ,1.0, ,1.84, 0.0 C m The final answer is D 1.aˆ 1.84aˆ 0.0 aˆ C m x y z Electrostatics -- Point Charges Slide 7 Force on a Charge 19

20 9/1/018 Force on a Point Charge E Force is most closely associated with electric field intensity. E F F Applied electric field may be present due to other charges, an applied voltage, a wave, etc. F E F force vector N Electrostatics -- Point Charges Slide 9 Example #4 What is the force on an electron located between two metal plates separated by 1 cm and 10 volts is applied? Solution Sketch the problem V E - e cm Calculate the electric field intensity E. V E d 10 V 0.01 m 1000 V m Calculate the force on the electron. F E C1000 V m N 160 an attonewtons Electrostatics -- Point Charges Slide 40 0

21 9/1/018 Coulomb s Law We have two charges, 1 and. F 1 at r 1 1 â1 R1 E at r Charge creates an electric flux D. r r D 4 r r Permittivity D The electric field intensity E is calculated from the electric flux density D. 1 E D Force on 1 due to is ˆ r r F a 4 R 4 r r Electric field E puts a force on charge 1. F E 1 1 Electrostatics -- Point Charges Slide 41 Summary of Direction of Force on 1 Due to + + E 1 F 1 E - F E F 1-1 E F 1 Electrostatics -- Point Charges Slide 4 1

22 9/1/018 Notes About Coulomb s Law Direction of force is along the line joining the two charges. Force is directly proportional to 1. Force is inversely proportional to R. Charges must be stationary for these equations to be valid (i.e. electrostatics). Electrostatics -- Point Charges Slide 4 Example #5 Calculate the attractive force between the electron and proton in a hydrogen atom. Solution Sketch the problem. r Charge: 1 Distance: r 5 pm C C Calculate the force F aˆ rr C C F m m a N 8 nn 8 ˆr aˆ r Electrostatics -- Point Charges Slide 44

23 9/1/018 Coulomb s Law for Multiple Charges The force on charge 0 due to charges 1 to N is r r F F a N N N 0 i 0 i i 0 0 ˆ 0i 0i i1 i1 4 R0i i1 4 ri r0 Electrostatics -- Point Charges Slide 45 Coulomb s Law for Multiple Charges The force on charge 0 due to charges 1 to N is r r F F a N N N 0 i 0 i i 0 0 ˆ 0i 0i i1 i1 4 R0i i1 4 ri r0 Electrostatics -- Point Charges Slide 46

24 9/1/018 Coulomb s Law for Multiple Charges The force on charge 0 due to charges 1 to N is r r F F a N N N 0 i 0 i i 0 0 ˆ 0i 0i i1 i1 4 R0i i1 4 ri r0 Electrostatics -- Point Charges Slide 47 Coulomb s Law for Multiple Charges The force on charge 0 due to charges 1 to N is r r F F a N N N 0 i 0 i i 0 0 ˆ 0i 0i i1 i1 4 R0i i1 4 ri r0 Electrostatics -- Point Charges Slide 48 4

25 9/1/018 Coulomb s Law for Multiple Charges The force on charge 0 due to charges 1 to N is r r F F a N N N 0 i 0 i i 0 0 ˆ 0i 0i i1 i1 4 R0i i1 4 ri r0 Electrostatics -- Point Charges Slide 49 Coulomb s Law for Multiple Charges The force on charge 0 due to charges 1 to N is r r F F a N N N 0 i 0 i i 0 0 ˆ 0i 0i i1 i1 4 R0i i1 4 ri r0 Electrostatics -- Point Charges Slide 50 5

26 9/1/018 Coulomb s Law for Multiple Charges The force on charge 0 due to charges 1 to N is r r F F a N N N 0 i 0 i i 0 0 ˆ 0i 0i i1 i1 4 R0i i1 4 ri r0 Electrostatics -- Point Charges Slide 51 6