Lecture 5 - Professor Li Gao PDF

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PHYS 220B ELECTRICITY AND MAGNETISM

Lecture #5 Chapter 22: Gauss’s Law 22.4 Applications of Gauss’s Law An electrostatic situation is defined as the one in which the charges have no net motion. In electrostatics, the electric field at every point within the material of a conductor must be zero, and any excess charge must reside on the conductor ’s surface. (If there is charge within the conductor, there must be non-zero electric field on the Gaussian surface around this charge; therefore, it is not possible.)

Gauss’s Law: determine field from charge; determine charge from field. 1. Field of a Charged Conducting Sphere: Example 22.5

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PHYS 220B ELECTRICITY AND MAGNETISM

2. Field of a Uniform Line Charge: Example 22.6

3. Field of an infinite plane sheet of charge: Example 22.7

4. Field between Oppositely Charged Parallel Conducting Plate

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5. Field of a Uniformly Charged Sphere

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6. Charge on a Hollow Sphere: Example 22.10

22.5 Charges on Conductors 1. In an electrostatic situation, in which there is no net motion of charge, the electric field at every point within a conductor is zero. (1) For a solid conductor, any excess charge is located entirely on the surface of the conductor. (2) If there is a cavity inside a conductor and there is no charge within the cavity, the net charge on the surface of the cavity is zero. (3) If there is a small body with a charge q inside a cavity within a conductor and the conductor is uncharged and is insulated from the charge q, there is a charge –q distributed on the surface of the cavity, a charge +q appears on the outer surface of the conductor. (4) If there is a small body with a charge q inside a cavity within a conductor with a charge 𝑞𝐶 , there is a charge –q distributed on the surface of the cavity, a charge +q appears on the outer surface of the conductor.

2. A Faraday cage (shield), which is an enclosure formed by conductive material or by a mesh of such material, blocks external electric fields. An external static electric field causes the electric charges within the cage’s conducting material to be distributed such that they cancel the field’s effect in the cage’s interior. This phenomenon is called electrostatic shielding and is used to protect electronic equipment from lightning strikes and electrostatic discharges.

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3. Field at the Surface of a Conductor At a point just outside any conductor with a surface charge density σ, the direction of 󰇍𝑬 is always perpendicular to the surface, which will be shown in Chapter 23. 𝐸⊥ 𝐴 =

𝜎𝐴 𝜀0

𝐸⊥ =

𝜎 𝜀0

So the field at the surface of a conductor:

Examples: 1. A solid metal sphere with radius 0.450 m carries a net charge of 0.250 nC. Find the magnitude of the electric field (a) at a point 0.100 m outside the surface of the sphere and (b) at a point inside the sphere, 0.100 m below the surface.

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2. A hollow, conducting sphere with an outer radius of 0.250 m and an inner radius of 0.200 m has a uniform surface charge density of +6.37× 10−6 C/m2 . A charge of -0.500 μC is now introduced into the cavity inside the sphere. (a) What is the new charge density on the outside of the sphere? (b) Calculate the strength of the electric field just outside the sphere. (c) What is the electric flux through a spherical surface just inside the inner surface of the sphere?

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3. An insulating sphere of radius 𝑅 = 0.160 m has uniform charge density 𝜌 = +7.20 × 10−9 C/m3 . A small object that can be treated as a point charge is released from rest just outside the surface of the sphere. The small object has positive charge 𝑞 = 3.40× 10−6 C. How much work does the electric field of the sphere do on the object as the object moves to a point very far from the sphere?

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Questions: 1. A conducting spherical shell, with inner radius a and outer radius b, has a positive point charge Q located at its center. The total charge on the shell is –3Q, and it is insulated from its surroundings. In the region a < r < b,

A. the electric field points radially outward. B. the electric field points radially inward. C. the electric field is zero. D. not enough information given to decide Answer: C 2. An uncharged conductor has a hollow cavity inside of it. Within this cavity there is a charge of +10 µC that does not touch the conductor. There are no other charges in the vicinity. Which statement about this conductor is true? A) The inner surface of the conductor carries a charge of -10 µC and its outer surface carries no excess charge. B) The inner and outer surfaces of the conductor each contain charges of -5 µC. C) The net electric field within the material of the conductor points away from the +10 µC charge. D) The outer surface of the conductor contains +10 µC of charge and the inner surface contains -10 µC. E) Both surfaces of the conductor carry no excess charge because the conductor is uncharged. Answer: D 3. A solid nonconducting sphere of radius R carries a uniform charge density throughout its volume. At a radial distance r1 = R/4 from the center, the electric field has a magnitude E0. What is the magnitude of the electric field at a radial distance r2 = 2R? A) E0/4 B) zero C) E0/2 D) E0 E) 2E0 Answer: D 4. A solid nonconducting sphere of radius R carries a charge Q distributed uniformly throughout its volume. At a certain distance r1 (r1 < R) from the center of the sphere, the 8/9

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electric field has magnitude E. If the same charge Q were distributed uniformly throughout a sphere of radius 2R, the magnitude of the electric field at the same distance r1 from the center would be equal to A) E/8. B) E/2. C) 2E. D) 8E. E) E. Answer: A 5. A hollow conducting spherical shell has radii of 0.80 m and 1.20 m, as shown in the figure. The sphere carries a net excess charge of -500 nC. A point charge of +300 nC is present at 9 2 the center. (k = 1/4πε0 = 8.99 × 10 N ∙ m /C) The radial component of the electric field at a point that is 1.50 m from the center is closest to

A) +1200 N/C. B) +2000 N/C. C) -800 N/C. D) -1600 N/C. E) -2000 N/C. Answer: C

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