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Gauss's Law Examples: Charged Shell, Infinite Plane, Long Wire

This review provides examples of Gauss's Law, including calculations of flux, charge enclosed, and electric field for symmetric charge distributions such as a charged spherical shell, an infinite plane, and a long straight wire.

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Gauss's Law Examples: Charged Shell, Infinite Plane, Long Wire

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  1. Gauss’s Law (II) • Examples: charged spherical shell, infinite plane, • long straight wire Review: For a closed surface S: Net outward flux through S

  2. Calculating E for symmetric charge distributions • Use symmetry to sketch the behaviour of E (possible only for very symmetric distributions) • Pick an imaginary “Gaussian surface” S • Calculate the flux through S, in terms of an unknown |E| • Calculate the charge enclosed by S from geometry • Relate 3) and 4) by Gauss’s Law, solve for |E|

  3. Example: Uniformly-Charged Thin Sheet E + + + + + + + + + + + Gaussian surface Find E due to an infinite charged sheet

  4. Solution

  5. Example: Infinite Line Charge (Long thin Wire) r + + + + + + + + + L Find E at distance r from the wire

  6. Solution

  7. Conductors (in Electrostatic Equilibrium) • inside a conductor. • just outside a conductor. • Any net charge on a conductor resides on the surface only.

  8. Example: Hollow Thin Spherical Shell + + Total charge Q + P1 + + r a + + r + + P2 + + + + Find E outside at P1 : (r > a): Find E inside at P2 : (r < a):

  9. Solution

  10. 1) If , charges would move!!! 2) If , charges would move!!! Consequences if E was not zero: E Ell + E=0 inside

  11. 3) inside, so any Gaussian surface inside the conductor encloses zero net charge. + + + + + + + + + + + + + + + + + + + + + + + + + + + + So net charge =0 Since E = 0 everywhere .

  12. + - - + - + conductor - + -

  13. dA1 dA2 Surface Charge Density on a Conductor E + + + + + E = 0 inside What is E at the surface of the conductor?

  14. Solution

  15. Example: • Metal ball of radius R1 has charge +Q • Metal shell of radius R2,and R3 has total charge -3Q ) • Find: charges on each surface. (and E) R3 +Q R1 R2 -3Q

  16. Solution

  17. Quick Recap At surface of conductor Conductor: Infinite Sheet of charge: Why is the lower value half of the upper?

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