Electric Fields in University Physics: Charges, Forces, and Fields

Lecture 7 Ch22. Finding the Electric Field I University Physics: Waves and Electricity Dr.-Ing. Erwin Sitompul http://zitompul.wordpress.com 2019

Homework 6: Charge Triangle 1. Three charges are arranged in an equilateral triangle with the side length d=5 cm. All three charges equal 2 nC. Find the vector of net force acting on charge at the top of the triangle. 2. Two point charges are located on an xy-plane as follows. The first charge, q1 = 4 C is located at ( 3,1). The second charge, q2 = 2 C is located at (1, 5). Determine the force vector acting on q1, i.e. F12. Erwin Sitompul University Physics: Wave and Electricity 7/2

Solution of Homework 6: Triangle of Charges = + F F F 1. top top, due to bottom left q q r top, due to bottom right q q r r = + 21 r 31 1 3 1 2 k k 2 2 21 31 9 9 (2 10 )(2 10 ) (5 10 ) (2 10 )(2 10 ) (5 10 ) = 8.99 10 + + 9 (0.5i 0.867j) 2 2 9 9 8.99 10 ( 0.5i + 9 0.867j) 2 2 5 = 2.494 10 j N = + 21 r cos60 i sin60 j d d 60 21 r = r + 0.5i cos120 i d 0.866j + 31 r 21 r = sin120 j d 31 31 r = r = + d 0.5i r 0.866j 5 10 = 60 60 120 = 2 m 21 31 Erwin Sitompul University Physics: Wave and Electricity 7/3

Solution of Homework 6: Triangle of Charges q q F k r r 6 9 (4 10 )(2 10 ) 8.99 10 ( 52) 1.383 10 ( 0.555i 0.832j) = + 2. = 21 1 2 12 2 21 6 = ( 0.555i + 0.832j) 2 3 7.676 10 i 1.151 10 j N = + 4 3 = r r r 21 1 2 3 1 = = + 4i 6j 1 5 r r = 21 r 21 21 4i + 6j = = + 0.555i 0.832j 52 Erwin Sitompul University Physics: Wave and Electricity 7/4

The Electric Field The Coulomb s law tells us how a charged particle interacts with another charged particle. The question now: Since the particles do not touch, how can one particle push or pull the other? How can there be such an action at a distance with no visible connection between the particles? The concept of Electric Field is introduced to explain this question. Erwin Sitompul University Physics: Wave and Electricity 7/5

The Electric Field The electric field is a vector field. It consists of a distribution of vectors, one for each point in the region around a charged object. We can define the electric field at some point, such at point P, by placing a positive charge q0, called a test charge. We then measure the electrostatic force F that acts on the test charge. The electric field E at point P is defined as: F E q = 0 The direction of force defines the direction of field Erwin Sitompul University Physics: Wave and Electricity 7/6

The Electric Field The positive test charge q0does not see the charged object. Instead, it feels the electric field produced by the charged object and gives response. The SI unit for the electric field is the newton per coulomb (N/C). Note: The field at point P existed both before and after the test charge was put there. Here, we assume that the presence of the test charge does not affect the charge distribution on the charged object. Erwin Sitompul University Physics: Wave and Electricity 7/7

Electric Field Lines In order to understand it better, we will try to visualize the electric field now. Michael Faraday introduced the idea of electric fields in the 19th century and thought of the space around a charged body as filled with electric field lines . The direction of the field lines indicate the direction of the electric force acting on a positive test charge. The density of the field lines is proportional to the magnitude of the field. The field strength is related to the number of lines that cross a certain unit area perpendicular to the field Erwin Sitompul University Physics: Wave and Electricity 7/8

Electric Field Lines The figure shows a positive test charge, placed near a sphere of uniform negative charge. The electrostatic force points toward the center of the sphere. The electric field vectors at all points are directed radially toward the sphere. The spreading of the field lines with distance from the sphere tells us that the magnitude of the electric field (field strength) decreases with distance from the sphere. Erwin Sitompul University Physics: Wave and Electricity 7/9

Electric Field Lines Electric field lines extend away from positive charge (where they originate) and towardnegative charge (where they terminate). Erwin Sitompul University Physics: Wave and Electricity 7/10

Electric Field Lines The figure below shows an infinitely large, nonconducting sheet (or plane) with a uniform distribution of positive charge on one side. Due to symmetry, some forces will cancel one another. The net electrostatic force on the positive test charge will be perpendicular to the sheet and point away from it. Erwin Sitompul University Physics: Wave and Electricity 7/11

The Electric Field Due to a Point Charge From Coulomb s law, the electrostatic force due to q1, acting on a positive test charge q0 is: r F k r 0 1 q q = 0 10 2 10 The electric field due to a point charge q1 is: F q q = = 0 r E 1 k 0 10 2 10 r 0 The field of a positive point charge is shown on the right, in vector form. The magnitude of the field depends only on the distance between the point charge (as the field source) and the location where the field is measured. Erwin Sitompul University Physics: Wave and Electricity 7/12

The Electric Field Due to a Point Charge Electric field is a vector quantity. Thus, the net, or resultant, electric field due to more than one point charge is the superposition of the field due to each charge. The net electric field at the position of the test charge, due to n point charges, is: E E E E E = + + + + q q k k r r 0,net 01 02 03 0n q q = + + + + 10 r 20 r 30 r r 3 n 1 2 k k n 2 2 2 2 30 r r 10 20 0 n n q = r m E k 0,net 0 m 2 r = 1 m 0 m Erwin Sitompul University Physics: Wave and Electricity 7/13

Checkpoint The figure below shows a proton p and an electron e on an x axis. What is the direction of the electric field due to the electron at: (a) Point S? (b) Point R? What is the direction of the net electric field at (c) Point R? (d) Point S? Rightward Rightward Leftward Leftward p and e have the same charge magnitude R and S are closer to e than to p. Erwin Sitompul University Physics: Wave and Electricity 7/14

Example 1 A point source q1 = 20 nC is located at S(1,4). Find the electric field E at P(5,1). All units are in SI. i 4j 5i + + Sr = + Pr = = = = + 4i 3j 5 j) (i = + = SP r r r r (5i 4j) 4i 3j P S j 2 2 5 (4) r r ( 3) SP = = = r 0.8i 0.6j SP SP SP q Field at the measurement point = 2 r S E k P SP SP r Source charge 9 (20 10 ) (5) q = (0.8i 0.6j) 9 8.99 10 = r 1 E k 2 0 10 2 10 r = 5.754i 4.315j N C Vector pointing from source charge to measurement point Erwin Sitompul University Physics: Wave and Electricity 7/15

Example 2 A point source q1 = 20 nC is located at S(1,4). Determine some points near S, where the magnitude of the electric field E is equal to 30 N/C. q = 2 r S E k P SP SP r q r = E E The magnitude of a vector is the scalar value of the vector itself = 2 r S E k P SP = r 1 The magnitude of a unit vector always equals 1 SP q = S E kr P 2 SP 9 (20 10 ) r 9 9(20 10 ) 30 The points near S must be 2.448 m away from S Where are they? = 9 30 8.99 10 = 2 8.99 10 SP r 2 SP = 2.448 SP r Erwin Sitompul University Physics: Wave and Electricity 7/16

Example 2 y Location of points where the magnitude of E is equal to 30 N/C P2 Some points near S with E = 30 N/C are: P1(3.448,4) P2(1,6.448) P3( 1.448,4) P4(1,1.552) q1 + P1 4 P3 r = 2.448 P4 x 1 Erwin Sitompul University Physics: Wave and Electricity 7/17

Checkpoint The figure here shows four situation in which charged particles are at equal distances from the origin. Rank the situations according to the magnitude of the net electric field at the origin, greatest first. All the same Be sure to know how and why Erwin Sitompul University Physics: Wave and Electricity 7/18

Example 3 Three particles with charges q1 = 2Q, q2 = 2Q, and q3 = 4Q, each with distance d from the origin, are shown in the figure below. What net electric field E is produced at the origin? = + q r + E E E E 1 2 3 q q = + + 10 r 20 r 30 r 3 1 2 k k k 2 2 2 20 r 30 r 10 = = = cos( 30 )i cos(150 )i cos(210 )i + + + sin( 30 )j sin(150 )j sin(210 )j 10 r r r d d d d d d 20 30 Erwin Sitompul University Physics: Wave and Electricity 7/19

Example 3 q q q = + + 10 r 20 r 30 r 3 1 2 E k k k 2 2 2 10 r 2 d 20 r 30 r 1 2 1 2 ( 2 ) d 1 2 1 2 ( 4 ) d 1 2 1 2 Q Q Q j j j = + + + 3i 3i 3i k k k 2 2 2 6.928 kQ 8 1 2 Q i N C = = 3 i N C kd 2 2 d E Erwin Sitompul University Physics: Wave and Electricity 7/20

A Point Charge in an Electric Field The electrostatic force F acting on a charged particle located in an external electric field E has the direction of E if the charge q of the particle is positive and has the opposite direction of E if negative. = F qE The motion of charged particle can now be analyzed by using the Newton s laws. The electrostatic force, if not in check, will accelerate a charged particle. Erwin Sitompul University Physics: Wave and Electricity 7/21

Checkpoint (a) In the figure, what is the direction of the electrostatic force on the electron due to the external field shown? Leftward y E (b) In which direction will the electron accelerate if it is moving parallel to the z axis before it encounters the external field? Leftward x e (c) If, instead, the electron is initially moving rightward, will its speed increase, decrease, or remain constant? Decrease The direction of force and field are the same on positive charge, but opposite on negative charge. The acceleration has the same direction as the force. Erwin Sitompul University Physics: Wave and Electricity 7/22

Measuring the Elementary Charge Robert A. Millikan in 1910-1913 devised an apparatus to measure the elementary charge e. When tiny oil drops are sprayed into chamber A, some become charged, either + or . They then drift to chamber C. If switch S is open, battery B has no electrical effect on chamber C. If switch S is closed, the battery causes the plates to be charged and set up a downward-directed electric field E in chamber C. Negatively charged drop will tend to drift upward, positively charged drop will drift downward. By timing the motion of oil drops, the effect of charge q can be determined, q = ne, where n is integer and e = 1.602 10 19 C. Millikan oil-drop apparatus Erwin Sitompul University Physics: Wave and Electricity 7/23

Pollination and Electrostatics The ability of a bee to transport pollen from flower to flower depends on the difference of electric charge existing the bee (positive), the pollen grain (neutral) and the stigma (negative, because electrically connected to the ground). Erwin Sitompul University Physics: Wave and Electricity 7/24

Ink-Jet Printing Ink-jet printer is an invention to meet the need for high-quality, high-speed printing. The figure below shows a negatively charged ink drop moving between two conducting plate. A uniform downward-directed electric field E has been set up. The drop is deflected upward and then strikes the paper at a position that is determined by the magnitudes of E and the charge q of the drop. Erwin Sitompul University Physics: Wave and Electricity 7/25

Example 4 An ink drop with a mass m of 1.3 10 10 kg and a negative charge of magnitude Q = 1.5 10 13 C enters the region between the plates of an ink-jet printer. The drop initially moves along the x axis with the speed vx = 18 m/s. The length L of each plate is 1.6 cm. The plates are charged and thus produce an electric field at all points between them. Assume that field E is downward directed, is uniform, and has a magnitude of 1.4 106 N/C. What is the vertical deflection of the drop at the far edge of the plates? (The gravitational force on the drop is small relative to the electrostatic force acting on the drop and can be neglected.) Erwin Sitompul University Physics: Wave and Electricity 7/26

Example 4 = = + 2 F QE ( 1.5 10 = 2.1 10 j N = y y v t a t 1 2 0 0 y y 6 )( 1.4 10 j) 13 = + 4 (0)(8.889 10 ) (1.615 10 )(8.889 10 ) 7 y direction 4 2 3 1 2 F m 2.1 10 j 1.3 10 1.615 10 jm s = 4 6.380 10 m = a = 0.638mm 7 = 10 Can you locate where the 0.638 mm is? = 3 2 y direction 2 x v 1.6 10 18 = = t = 8.889 10 s 4 x Erwin Sitompul University Physics: Wave and Electricity 7/27

Homework 7: Three Particles Three particles are fixed in place and have charges q1 = q2 = +p and q3 = +2p. Distance a = 6 m. What are the magnitude and direction of the net electric field at point P due to the particles? 1.602 10 = + 1.602 10 = 19 C C p e 19 Erwin Sitompul University Physics: Wave and Electricity 7/28

Homework 7A: Two Particles 1. The figure below shows two charged particles on an x axis, q= 3.2 10 19 C at x= 3 m and q= 4.8 10 19 C at x= 3 m. What are the (a) magnitude and (b) direction of the net electric field produced at point P at y= 4 m? 2. (a) A force of 8.4 N directed to the south is applied on a 8.8 C charge. Calculate the strength and direction of the electric field at this point. (b) What is the magnitude of an electric field at a point in space where a proton (m=1.673 10 27 kg) experiences an acceleration of one million g ? Erwin Sitompul University Physics: Wave and Electricity 7/29

Homework 7B: Four Particles 1. Three charges are placed on three corners of a square. Each side of the square is 30.0 cm. (a) Compute E at the fourth corner in vector notation. (b) What would be the force on a 6.00 C charge placed at the vacant corner? Erwin Sitompul University Physics: Wave and Electricity 7/30