1/16/07184 Lecture 51 PHY 184 Spring 2007 Lecture 5 Title: Electric Field Examples

1/16/07184 Lecture 52AnnouncementsAnnouncements  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.

1/16/07184 Lecture 53 Review - The Electric Field  A charge creates an electric field around itself and the other charge feels that field. Test charge: point object with a very small positive charge so that it does not modify the original 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 q +

1/16/07184 Lecture 54 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

1/16/07184 Lecture 55 Electric Field from a Point Charge  Suppose we have two charges, q and q 0, separated by a distance r. The electric force between the two charges is  We can consider q 0 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.

1/16/07184 Lecture 56 Electric Field from 4 Point Charges (1)  Four charges q 1 =10 nC, q 2 =-20 nC, q 3 =20 nC and q 4 =-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.

1/16/07184 Lecture 57 Electric Field from 4 Point Charges (2) x component (at 0) E 2x is positive! E 3x is positive! q1 = 10 nC, q2 = -20 nC q3 = 20 nC, q4 = -10 nC

1/16/07184 Lecture 58 Electric Field from 4 Point Charges (3) y component (at 0) E 1y is negative! E 4y is negative! q1 = 10 nC, q2 = -20 nC q3 = 20 nC, q4 = -10 nC

1/16/07184 Lecture 59 Electric Field from 4 Point Charges (4) E at the center pt.

1/16/07184 Lecture 510 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

1/16/07184 Lecture 511 Electric Field from Three Point Charges (2)  The electric field at P due to q 1 is  The electric field at P due to q 3 is  The electric field at P due to q 2 is Note: tan  = a/b

1/16/07184 Lecture 512  Now we add the components, to obtain  Magnitude of E  Direction of E Electric Field from Three Point Charges (3)

1/16/07184 Lecture 513 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

1/16/07184 Lecture 514 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.

1/16/07184 Lecture 515 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

1/16/07184 Lecture 516 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

1/16/07184 Lecture 517 Electric Field from an Electric Dipole (5)  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) Taylor series

1/16/07184 Lecture 518 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

1/16/07184 Lecture 519 Functional Dependence E(x) 1/4 1/8 E(x) E(x)= Point charge Dipole

1/16/07184 Lecture 520 Electric Dipole Moment of a Water Molecule  Chemistry reminder - the H 2 O molecule The distribution of electric charge in a H 2 O 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."

1/16/07184 Lecture 521 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?

1/16/07184 Lecture 522 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 6.2x C  m. (The assumed charge distribution not precise.)

1/16/07184 Lecture 523 Demo - Polarization

1/16/07184 Lecture 524 Math Reminder (1) a a d a b c x y

1/16/07184 Lecture 525 Math Reminder (2) Binomial theorem : (Taylor series) In our case: x=d/2x, n=-2 and x=-d/2x, n=-2