PHY 184

1. PHY 184 Spring 2007 Lecture 5 Title: Electric Field Examples 184 Lecture 5

2. Announcements • Homework Set 1 is done. The average score was 9.6/10. • Homework Set 2 is open - we will open Homework Sets now already on Thursday. • Helproom coverage is posted in LON-CAPA. • Honors option work in the SLC: • Please sign up for time slots! Use the sign-up sheet after class. 184 Lecture 5

3. Review - The Electric Field + + Test charge q • A charge creates an electric field around itself and the other charge feels that field. The electric field at a specified point: place a positive test charge q at the point and measure the electrostatic force that acts on the test charge, Test charge: point object with a very small positive charge so that it does not modify the original field 184 Lecture 5

4. Review -Field Lines from a Point Charge • The electric field lines from a point charge extend out radially. • For a positive point charge, the field lines point outward • Terminate at infinity • For a negative charge, the field lines point inward • Originate at infinity 184 Lecture 5

5. Electric Field from a Point Charge • Suppose we have two charges, q and q0, separated by a distance r. The electric force between the two charges is • We can consider q0 to be a test charge, and determine the electric field from charge q as • The electric field is a vector, so to add electric fields we must add the components separately. 184 Lecture 5

6. Electric Field from 4 Point Charges (1) • Four charges q1=10 nC, q2=-20 nC, q3=20 nC and q4=-10 nC form a square of edge length 5 cm. What electric field do the particles produce at the square center? • Idea: Use the superposition principle. • Step 1: Choose your coordinate system and stick with it! • Step 2: Look at the x and y coordinates separately. 184 Lecture 5

7. Electric Field from 4 Point Charges (2) q1 = 10 nC, q2 = -20 nC q3 = 20 nC, q4 = -10 nC x component (at 0) E2x is positive! E3x is positive! 184 Lecture 5

8. Electric Field from 4 Point Charges (3) q1 = 10 nC, q2 = -20 nC q3 = 20 nC, q4 = -10 nC y component (at 0) E1y is negative! E4y is negative! 184 Lecture 5

9. Electric Field from 4 Point Charges (4) E at the center pt. 184 Lecture 5

10. Electric Field from Three Point Charges • Consider three charges • The three charges are placed at • What is the electric field at point P? P: (b,a) a = 8.0 m ; b = 6.0 m 184 Lecture 5

11. Electric Field from Three Point Charges (2) • The electric field at P due to q1 is • The electric field at P due to q3 is • The electric field at P due to q2 is Note: tanq = a/b 184 Lecture 5

12. Electric Field from Three Point Charges (3) • Now we add the components, to obtain • Magnitude of E • Direction of E 184 Lecture 5

13. Electric Field from an Electric Dipole • A system of two oppositely charged point particles is called an electric dipole. • The vector sum of the electric field from the two charges gives the electric field of the dipole (superposition principle). • We have shown the electric field lines from a dipole 184 Lecture 5

14. Electric Field from an Electric Dipole (2) • Expression for the electric field of a dipole along a line including both charges … • We will derive a general expression good anywhere along the dashed line and then get an expression for the electric field a long distance from the dipole. 184 Lecture 5

15. Electric Field from an Electric Dipole (3) • Two charges on the x-axis a distance d apart • Put -q at x = -d/2 • Put +q at x = +d/2 • Calculate the electric field at a point P a distance x from the origin 184 Lecture 5

16. Electric Field from an Electric Dipole (4) • Principle of superposition:The electric field at any point x is the sum of the electric fields from +q and -q • Replacing r+ and r- we get 184 Lecture 5

17. Electric Field from an Electric Dipole (5) Taylor series • This equation gives the electric field everywhere on the x-axis (except for x = d/2) • Let’s look at this equation far away along the positive x-axis (x >> d) 184 Lecture 5

18. Definition of Electric Dipole Moment • We define the vector electric dipole moment as a vector that points from the negative charge to the positive charge • p is the magnitude of the dipole moment • q is the magnitude of one of the opposite charges • d is the distance between the charges • Using this definition we can write the electric field far away from an electric dipole as 184 Lecture 5

19. Functional Dependence E(x) Point charge E(x)= E(x)= Dipole E(x) 1/4 1/8 184 Lecture 5

20. Electric Dipole Moment of a Water Molecule • Chemistry reminder - the H2O molecule The distribution of electric charge in a H2O molecule is non-uniform. The more electronegative oxygen atom attracts electrons from the hydrogen atoms. Thus, the oxygen atom acquires a partial negative charge and the hydrogen atoms acquire a partial positive charge. The water molecule is "polarized." 184 Lecture 5

21. Example - Electric Dipole Moment of Water • Suppose we approximate the water molecules as two positive charges located at the center of the hydrogen atoms and two negative charges located at the center of the oxygen atom. What is the electric dipole moment of a water molecule? 184 Lecture 5

22. Electric Dipole Moment of Water (2) Our result for the electric dipole moment of water is then This oversimplified result is comparable to the measured value of 6.2x10-30 C  m. (The assumed charge distribution not precise.) 184 Lecture 5

23. Demo - Polarization 184 Lecture 5

24. Math Reminder (1) c y a x b d a a 184 Lecture 5

25. Math Reminder (2) Binomial theorem : (Taylor series) In our case: x=d/2x, n=-2 and x=-d/2x, n=-2 184 Lecture 5