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Understanding Gauss's Law for Different Symmetries in Electric Fields

This lecture discusses Gauss's Law and its application for spherical, line, and plane symmetries in electric fields. It covers topics such as conductors in electric fields and Coulomb's law. Examples and checkpoints are provided to enhance understanding.

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Understanding Gauss's Law for Different Symmetries in Electric Fields

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  1. Last Lecture Gauss’s law Using Gauss’s law for: spherical symmetry This lecture Using Gauss’s law for: line symmetry plane symmetry Conductors in electric fields

  2. Coulomb’s law tutorial: Consider two positively charged particles, one of charge q0 (particle 0) fixed at the origin, and another of charge q1 (particle 1) fixed on the y-axis at (0, d1, 0). What is the net force Fon particle 0 due to particle 1? Express your answer (a vector) using any or all of k, q0, q1, d1, x-hat, y-hat, and z-hat. Answer: F = Example of wrong answer: ()

  3. Part B What is the new net force on particle 0, from particle 1 and particle 2? Answer: F = Here is an example of a wrong answer - just a careless mistake. This one presumably meant to put a bracket around the two terms, but didn’t.

  4. Part D What is the net force on particle 0 due solely to this charge 3? Answer: F = The answer here was close but still missing a factor of ½. The message is that accuracy matters.

  5. A B C D E These are two-dimensional cross sections through three dimensional closed spheres and a cube. Which of them has the largest flux through surfaces A to E? Which has the smallest flux? Answers: ΦB= ΦE > ΦA= ΦC= ΦD The flux through a CLOSED surface depends only on the amount of enclosed charge, not the size or shape of the surface.

  6. Example 21.2 Field of a hollow spherical sphere From Gauss’s law, E = 0 inside shell Example 21.3 Field of a point charge within a shell Done on board Read TIP: SYMMETRY MATTERS!

  7. (line of charge) Line symmetry Example 21.4 Field of a line of charge A section of an infinitely long wire with a uniform linear charge density, . Find an expression for E at distance r from axis of wire.

  8. (line of charge) Applies outside any cylindrical charge distribution Applying Gauss’ Law: cylindrical symmetry (Compare this result with that obtained using Coulomb’s law in Example 20.7, when wire is infinitely long.)

  9. Plane symmetry Example 21.6 Field of an infinite plane sheet of charge p 357

  10. Gaussian surface (cylinder) (sheet of charge) Applying Gauss’ Law: planar symmetry A thin, infinite, nonconducting sheet with uniform surface charge density Find E at distance r from sheet.

  11. Electric field due to plane of charge is By integration methods: • which reduces to above formula for very large R • (see problem 20.69)

  12. CHECKPOINT: The figure shows two large parallel, nonconducting sheets with identical (positive) volume charge density. Rank the four labelled points according to the magnitude of the net electric field there, greatest first. A C D B Answer: C, D equal B A

  13. 21.5 Fields of arbitrary charge distributions

  14. CHECKPOINT: There is a certain net flux I through a Gaussian sphere of radius r enclosing an isolated charged particle. Suppose the Gaussian surface is changed to • (a) a larger Gaussian sphere, • (b) a Gaussian cube with edge length equal to r, and • (c) a Gaussian cube with edge length 2r. • In each case is the net flux through the new Gaussian surface • greater than • less than, or • equal to I ? Answers: all equal as charge enclosed is the same

  15. When we must use Coulomb - no symmetry! • Find E at point P due to a finite line charge • Charge Q is uniformly distributed on a straight line segment of length L. • Choose axes and draw diagram • Write expression for |dE| due to dq • Find dEy • Integrate to find Ey • (what are the limits?) • Substitute trigonometric formulae • Repeat for Ex

  16. True A • True or False? • If the net electric flux out of a closed surface is zero, the electric field must be zero everywhere on the surface. • If the net electric flux out of a closed surface is zero, the charge density must be zero everywhere inside the surface. • The electric field is zero everywhere within the material of a conductor in electrostatic equilibrium. • The tangential component of the electric field is zero at all points just outside the surface of a conductor in electrostatic equilibrium. • The normal component of the electric field is the same at all points just outside the surface of a conductor in electrostatic equilibrium. False F False False True True False

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