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Understanding Electric Fields from Charge Ring and Dipoles: Concepts and Applications

This article explores the electric field generated by a ring of charge and dipoles, emphasizing the symmetry effects in electric fields and the significance of linear charge density. It describes the motion of a particle near the charged ring and its directional movement, alongside an analysis of dipole behavior in electric fields, specifically addressing force, torque, and conditions for equilibrium. The article also delves into electric flux concepts and Gauss's Law, illustrating applications with practical problems, including flux through surfaces surrounding charges.

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Understanding Electric Fields from Charge Ring and Dipoles: Concepts and Applications

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  1. Announcements

  2. Electric Field from a Ring of Charge Consider symmetry: all perpendicular fields cancel, only parallel components (cos q) r z R Linear charge density 

  3. Electric Field from a Ring of Charge r z R Linear charge density 

  4. Electric Field Suppose I put a charge close to the ring: Describe the motion of the particle: In what direction does the particle move? Does the particle move at a constant velocity ? Where does the particle stop ? r z q R Linear charge density 

  5. q2 = -1 C 10 cm 5 cm q1 = +1 C Dipoles • A dipole is a postive and negative charge separated by a distance d • Commonly found in molecules! Though the distances and charges are much smaller! Dipole moment is qd. It is a vector!

  6. E-field from a Dipole: Limits If z is large, ie, far away from the dipole The field from a dipole weakens as one goes further away faster than a point charge.

  7. q2 = -1 C q1 = +1 C E-field from a Dipole Dipole Axis d/2 What is the field along the dipole axis? d/2 Field from Point charge A dipole is just two point charges in a specific arrangment.

  8. q2 = -1 C q1 = +1 C E-field from a Dipole II Field from 2 Point charges Dipole Axis z Field along the dipole axis d/2 d/2 Some algebra

  9. Dipoles in an field - + In a uniform field, the force on each end is equal but opposite; no net force [as the charges are connected!] In a nonuniform field, the force on each end is not equal but opposite; So a net force

  10. Dipoles in an field In a uniform field, the force on each end is equal but opposite; no net force [as the charges are connected!] - + There can be a torque!! If the dipole moment makes an angle with the field, t=dFsinq, where F is the force on 1 charge, and so t=dqEsinq=pEsinq What is the lever

  11. Dipoles

  12. Electric Flux • Electric Flux is the amount of electric field flowing through a surface • When electric field is at an angle, only the part perpendicular to the surface counts E • Multiply by cos  En  • For a non-constant electric field, or a curvy surface, you have to integrate over the surface E = EnA= EA cos  • Usually you can pick your surface so that the integration doesn’t need to be done given a constant field

  13. R Electric Flux • What is electric flux through surface surrounding a charge q? charge q Answer is always 4keq

  14. Gauss’s Law charge q • Flux out of an enclosed region depends only on total charge inside A positive charge q is set down outside a sphere. Qualitatively, what is the total electric flux out of the sphere as a consequence? A) Positive B) Negative C) Zero D) It is impossible to tell from the given information

  15. R Gauss’ Law and Coulumb’s • Suppose we had • measured the flux as: • From Gauss’ law: So Gauss’ law implies Coulomb’s law charge q • What if we lived in a Universe with a different number of physical dimensions?

  16. Gauss’s Law charge q’’ charge q’ charge q

  17. Gauss’s Law charge q charge 2q charge -2q charge -q How do we draw surfaces to contain the +2q charge and have flux?: Zero ? +3q/e0? -2q/e0 ? IMPOSSIBLE

  18. Example • q/eo D) 2q/eo • -q/eo • 0 Figure 24-29. • What is the flux through the first surface? • What is the flux through the second surface? • What is the flux through the third surface? • What is the flux through the fourthsurface? • What is the flux through the fifth surface?

  19. Practive Problem I A cube with 1.40 m edges is oriented as shown in the figure • Suppose there is a charge situated in the middle of • the cube. • What is the magnitude of the flux through the whole cube? • What is the magnitude of the flux through any one side? • q/eo D) q/6eo • q/4eo • 0

  20. Practice Problem II A cube with 1.40 m edges is oriented as shown in the figure • Suppose the cube sits in a uniform electric field of 10i ? • What is the magnitude of the flux through the whole cube? • What is the magnitude of the flux through the top side? • How many sides have nonzero flux? • q/eo D) q/6eo • q/4eo • 0 • 2 D) 1 • 4 • 0

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