02. Electric Field II

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1 Universit of Rhode Island PHY 24: Elementar Phsics II Phsics Course Materials Electric Field II Gerhard Müller Universit of Rhode Island, Creative Commons License This work is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 4. License. Follow this and additional works at: Abstract Part two of course materials for Elementar Phsics II (PHY 24), taught b Gerhard Müller at the Universit of Rhode Island. Documents will be updated periodicall as more entries become presentable. Some of the slides contain figures from the tetbook, Paul A. Tipler and Gene Mosca. Phsics for Scientists and Engineers, 5 th /6 th editions. The copright to these figures is owned b W.H. Freeman. We acknowledge permission from W.H. Freeman to use them on this course web page. The tetbook figures are not to be used or copied for an purpose outside this class without direct permission from W.H. Freeman. Recommended Citation Müller, Gerhard, "2. Electric Field II" (215). PHY 24: Elementar Phsics II. Paper This Course Material is brought to ou for free and open access b the Phsics Course Materials at DigitalCommons@URI. It has been accepted for inclusion in PHY 24: Elementar Phsics II b an authorized administrator of DigitalCommons@URI. For more information, please contact digitalcommons@etal.uri.edu.

2 Particle in Uniform Electric or Gravitational Field particle charge mass electron q e = e m e = kg proton q p = +e m p = kg neutron q n = m n = kg Elementar charge: e = C. Electric field equation of motion: F = m a force law: F = q E acceleration: a = (q/m) E E q q n = F e p F p F n = q e Gravitational field equation of motion: F = m a g m m m p n e force law: F = m g acceleration: a = g F p F n F e 1/9/215 [tsl24 1/15]

3 Projectile Motion in Electric Field electrostatic force: F = F = ee equation of motion: acceleration: a = velocit: v (t) = v cos θ F = me a a = e m e E a v (t) = v sin θ at position: (t) = v [cos θ]t (t) = v [sin θ]t 1 2 at2 height: h = v2 2a sin2 θ range: R = v2 a sin(2θ) m e qe = e v θ R h E 1/9/215 [tsl26 2/15]

4 Cathode Ra Tube 1/9/215 [tsl419 3/15]

5 Particle Projected Perpendicular to Uniform Electric Field A charged particle (m = 3kg, q = 1µC) is launched at t = with initial speed v = 2m/s in an electric field of magnitude E = N/C as shown. m q E 1m 1m v (a) Find the position of the particle at t 1 = 3s. (b) B what angle does the velocit vector turn between t = and t 1 = 3s? 1/9/215 [tsl27 4/15]

6 Particles Accelerated b Uniform Electric Field A uniform electric field E = N/C eists in the bo. (a) A charged particle of mass m 1 = kg is released from rest at = 3cm, =. It eits the bo at = 3cm, = 6cm after a time t 1 = s. Find the charge q 1. (b) A second charged particle of mass m 2 = kg is projected from position =, = 3cm with initial speed v = m/s. It eits the bo at = 3.9cm, = 6cm. Find the charge q 2. 6 [cm] 4 q 2 E 2 v 2 q [cm] 1/9/215 [tsl28 5/15]

7 Action and Reaction due to Coulomb Interaction Two particles with masses m 1, m 2 and charges q 1, q 2 are released from rest a distance r apart. We consider the following four distinct configurations: (a) m 1 = 1kg, m 2 = 1kg, q 1 = 1C, q 2 = 1C (b) m 1 = 1kg, m 2 = 1kg, q 1 = 1C, q 2 = 2C (c) m 1 = 1kg, m 2 = 2kg, q 1 = 1C, q 2 = 1C (d) m 1 = 1kg, m 2 = 2kg, q 1 = 1C, q 2 = 2C m 1 r q 1 m 2 q 2 Anwer the following questions for each configuration: (1) Is the force eperienced b particle 1 smaller than or equal to or larger than the force eperienced b particle 2? (2) Is the acceleration of particle 1 smaller than or equal to or larger than the acceleration of particle 2? 1/9/215 [tsl29 6/15]

8 Particle in Uniform Electric and Gravitational Field (1) A proton, a neutron, and an electron are dropped from rest in a vertical gravitational field g and in a horizontal electric field E as shown. Both fields are uniform. p n e g E h (a) Which particle travels the shortest distance? (b) Which particle travels the longest distance? (c) Which particle travels the shortest time? (d) Which particle reaches the highest speed? 1/9/215 [tsl25 7/15]

9 Particle in Uniform Electric and Gravitational Field (2) A proton, a neutron, and an electron are dropped from rest in a vertical gravitational field g and in a horizontal electric field E as shown. Both fields are uniform. g E (a) Which particle travels the shortest distance? (b) Which particle travels in a straight line? (c) Which particle travels the shortest time? (d) Which particle reaches the highest speed? e n p h h 1/9/215 [tsl42 8/15]

10 Is the Faster also the Quicker? Charged particles 1 and 2 are released from rest in a uniform electric field. (a) Which particle moves faster when it hits the wall? (b) Which particle reaches the wall more quickl? d = 2m 1 m = 1kg 1 q = 2C 1 E = 4N/C d = 1m 2 m = 1kg 2 q = 1C 2 1/9/215 [tsl372 9/15]

11 Intermediate Eam I: Problem #3 (Spring 6) Consider a region of uniform electric field as shown. A charged particle is projected at time t = with initial velocit as shown. Ignore gravit. (a) Find the components a and a of the acceleration at time t =. (b) Find the components v and v of the velocit at time t =. (c) Find the components v and v of the velocit at time t = 1.2s. (d) Find the components and of the position at time t = 1.2s. m =3g q = 6mC v = 2m/s 1/9/215 [tsl35 1/15]

12 Intermediate Eam I: Problem #3 (Spring 6) Consider a region of uniform electric field as shown. A charged particle is projected at time t = with initial velocit as shown. Ignore gravit. (a) Find the components a and a of the acceleration at time t =. (b) Find the components v and v of the velocit at time t =. (c) Find the components v and v of the velocit at time t = 1.2s. (d) Find the components and of the position at time t = 1.2s. Solution: (a) a = q m E = C kg (5N/C) = 1m/s2, a =. m =3g q = 6mC v = 2m/s 1/9/215 [tsl35 1/15]

13 Intermediate Eam I: Problem #3 (Spring 6) Consider a region of uniform electric field as shown. A charged particle is projected at time t = with initial velocit as shown. Ignore gravit. (a) Find the components a and a of the acceleration at time t =. (b) Find the components v and v of the velocit at time t =. (c) Find the components v and v of the velocit at time t = 1.2s. (d) Find the components and of the position at time t = 1.2s. Solution: (a) a = q m E = C kg (5N/C) = 1m/s2, a =. (b) v =, v = v = 2m/s. m =3g q = 6mC v = 2m/s 1/9/215 [tsl35 1/15]

14 Intermediate Eam I: Problem #3 (Spring 6) Consider a region of uniform electric field as shown. A charged particle is projected at time t = with initial velocit as shown. Ignore gravit. (a) Find the components a and a of the acceleration at time t =. (b) Find the components v and v of the velocit at time t =. (c) Find the components v and v of the velocit at time t = 1.2s. (d) Find the components and of the position at time t = 1.2s. Solution: (a) a = q m E = C kg (5N/C) = 1m/s2, a =. (b) v =, v = v = 2m/s. (c) v = a t = (1m/s 2 )(1.2s) = 12m/s, v = v = 2m/s. m =3g q = 6mC v = 2m/s 1/9/215 [tsl35 1/15]

15 Intermediate Eam I: Problem #3 (Spring 6) Consider a region of uniform electric field as shown. A charged particle is projected at time t = with initial velocit as shown. Ignore gravit. (a) Find the components a and a of the acceleration at time t =. (b) Find the components v and v of the velocit at time t =. (c) Find the components v and v of the velocit at time t = 1.2s. (d) Find the components and of the position at time t = 1.2s. Solution: (a) a = q m E = C kg (5N/C) = 1m/s2, a =. (b) v =, v = v = 2m/s. (c) v = a t = (1m/s 2 )(1.2s) = 12m/s, v = v = 2m/s. m =3g q = 6mC v = 2m/s (d) = 1 2 a t 2 =.5(1m/s 2 )(1.2s) 2 = 7.2m, = v t = (2m/s)(1.2s) = 2.4m. 1/9/215 [tsl35 1/15]

16 Unit Eam I: Problem #3 (Spring 7) Consider two regions of uniform electric field as shown. Charged particles of mass m = 2kg and charge q = 1C are projected at time t = with initial velocities as shown. Both particles will hit the screen eventuall. Ignore gravit. (a) At what time t 1 does the particle in region (1) hit the screen? (b) At what height 1 does the particle in region (1) hit the screen? (c) At what time t 2 does the particle in region (2) hit the screen? (d) At what height 2 does the particle in region (2) hit the screen? v = 2m/s (1) (2) screen v = 2m/s screen 8m 8m 1/9/215 [tsl361 11/15]

17 Unit Eam I: Problem #3 (Spring 7) Consider two regions of uniform electric field as shown. Charged particles of mass m = 2kg and charge q = 1C are projected at time t = with initial velocities as shown. Both particles will hit the screen eventuall. Ignore gravit. (a) At what time t 1 does the particle in region (1) hit the screen? (b) At what height 1 does the particle in region (1) hit the screen? (c) At what time t 2 does the particle in region (2) hit the screen? (d) At what height 2 does the particle in region (2) hit the screen? Solution: (a) 1 = 1 2 at2 1 with a = q m E = 2.5m/s2, 1 = 8m t 1 = 2.53s. v = 2m/s (1) (2) screen v = 2m/s screen 8m 8m 1/9/215 [tsl361 11/15]

18 Unit Eam I: Problem #3 (Spring 7) Consider two regions of uniform electric field as shown. Charged particles of mass m = 2kg and charge q = 1C are projected at time t = with initial velocities as shown. Both particles will hit the screen eventuall. Ignore gravit. (a) At what time t 1 does the particle in region (1) hit the screen? (b) At what height 1 does the particle in region (1) hit the screen? (c) At what time t 2 does the particle in region (2) hit the screen? (d) At what height 2 does the particle in region (2) hit the screen? Solution: (a) 1 = 1 2 at2 1 with a = q m E = 2.5m/s2, 1 = 8m t 1 = 2.53s. (b) 1 = v t 1 = 5.6m. v = 2m/s (1) (2) screen v = 2m/s screen 8m 8m 1/9/215 [tsl361 11/15]

19 Unit Eam I: Problem #3 (Spring 7) Consider two regions of uniform electric field as shown. Charged particles of mass m = 2kg and charge q = 1C are projected at time t = with initial velocities as shown. Both particles will hit the screen eventuall. Ignore gravit. (a) At what time t 1 does the particle in region (1) hit the screen? (b) At what height 1 does the particle in region (1) hit the screen? (c) At what time t 2 does the particle in region (2) hit the screen? (d) At what height 2 does the particle in region (2) hit the screen? Solution: (a) 1 = 1 2 at2 1 with a = q m E = 2.5m/s2, 1 = 8m t 1 = 2.53s. (b) 1 = v t 1 = 5.6m. (c) 2 = v t 2 t 2 = 8m 2m/s = 4s. v = 2m/s (1) (2) 8m screen v = 2m/s 8m screen 1/9/215 [tsl361 11/15]

20 Unit Eam I: Problem #3 (Spring 7) Consider two regions of uniform electric field as shown. Charged particles of mass m = 2kg and charge q = 1C are projected at time t = with initial velocities as shown. Both particles will hit the screen eventuall. Ignore gravit. (a) At what time t 1 does the particle in region (1) hit the screen? (b) At what height 1 does the particle in region (1) hit the screen? (c) At what time t 2 does the particle in region (2) hit the screen? (d) At what height 2 does the particle in region (2) hit the screen? Solution: (a) 1 = 1 2 at2 1 with a = q m E = 2.5m/s2, 1 = 8m t 1 = 2.53s. (b) 1 = v t 1 = 5.6m. (c) 2 = v t 2 (d) 2 = 1 2 at2 2 = 2m. t 2 = 8m 2m/s = 4s. v = 2m/s (1) (2) 8m screen v = 2m/s 8m screen 1/9/215 [tsl361 11/15]

21 Electric Dipole Field L E E + q +q p E =» kq ( L/2) 2 kq ( + L/2) 2 ( + L/2) 2 = kq ( L/2) 2 ( L/2) 2 ( + L/2) 2 2kqL 3 = 2kp 3 (for L) = 2kqL ( 2 L 2 /4) 2 Electric dipole moment: p = q L Note the more rapid deca of the electric field with distance from an electric dipole ( r 3 ) than from an electric point charge ( r 2 ). The dipolar field is not radial. 1/9/215 [tsl23 12/15]

22 Water Molecule 1/9/215 [tsl51 13/15]

23 Force and Torque on Electric Dipole The net force on an electric dipole in a uniform electric field vanishes. However, this dipole eperiences a torque τ = p L that tends to align the vector p with the vector E. Now consider an electric dipole that is alread aligned (locall) with a nonuniform electric field. This dipole eperiences a net force that is alwas in the direction where the field has the steepest increase. 1/9/215 [tsl328 14/15]

24 Electric Quadrupole Field L = L E E + +q 2q +q p p 2 1 E = = kq 2 kq ( L) 2 + kq ( + L) 2 + k( 2q) kqL2 4 2L + 3L2 2 + = 3kQ 4 (for L) = kq 2 1 " 1 L 2L + 3L2 2 q « L «# 2 2 «2 +q Electric quadrupole moment: Q = 2qL 2 Different quadrupole configuration: L L +q q. 1/9/215 [tsl327 15/15]

Particle in Uniform Electric or Gravitational Field

Particle in Uniform Electric or Gravitational Field particle charge mass electron q e = e m e = 9.19 1 31 kg proton q p = +e m p = 1.673 1 27 kg neutron q n = m n = 1.675 1 27 kg Elementar charge: e =