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Planar Symmetry: Infinite Sheet of Charge Non-Conducting Sheet s+ + + + + + + 1/26/2016 4 Example: Two Parallel Non-Conducting Sheets s+ + + + + + + + + + + + + 1/26/2016 s- - In this example, assume s+ > s- Find the electric field to the left of the sheets, between the sheets and to the right of the sheets. 5 Example: Gauss’ Law: Spherical Symmetry A spherically symmetric charge distribution has a charge density given by =a/r where a is a constant. Find the electric field as a function of r. o E dA qenc 2 0 0 o Er 2 d sin d o E 4 r 2 q 2 2 r 4 ar o E 4 r 2 q 4 r 2 dr a 4 dr 2 0 0 r r r a E 2 o 1/26/2016 Figure from: http://www.oceanopticsbook.info/view/light_and_radiometry/geometry 6 From Lecture #1: Conductors versus Insulators • Insulators: material in which electric charges are “frozen” in place. • Conductors: material in which electric charges can move around “freely. 1/26/2016 7 Conductors If an excess charge is placed on an isolated conductor, that amount of charge will move entirely to the surface of the conductor. None of the excess charge will be found within the body of the conductor. 1/26/2016 8 Conductors Isolator Conductor with a Cavity 1/26/2016 9 Gauss’ Law: Spherical Symmetry Demo: 5A-13 No Internal Field Electric Field inside and outside a shell of uniform charge distribution 1/26/2016 10 Conductors: Shielding from an Electrical Field • Electric field lines for an oppositely charged metal cylinder and metal plate. Note that: 1. Electric field lines are perpendicular to the conductors. 2. There are no electric field lines inside the cylinder. 1/26/2016 11 Conductors: Shielding from an Electrical Field student sensor sparks screened cage 1/26/2016 Van de Graaff generator 12 Conductors DEMO: 5A-12 Charge within a Conductor 1/26/2016 13 Conductors Charge Distribution on a Conductor 1/26/2016 http://cnx.org/content/m42317/latest/Figure_19_07_07a.jpg 14 Example: -100e A ball of charge -50e lies at the center of a hollow spherical metal shell that has a net charge -100e. -50e (1) What is the charge on the shell’s inner surface? (a) -50e 1/26/2016 (b) 0 (c ) +50e 15 Example: -100e A ball of charge -50e lies at the center of a hollow spherical metal shell that has a net charge -100e. -50e (2) What is the charge on the shell’s outer surface? (a) -150e 1/26/2016 (b) -50e (c) +100e 16 Question 1. • A solid conducting sphere is concentric with a thin conducting shell, as shown. • The inner sphere carries a charge Q1, and the spherical shell carries a net charge Q2, such that Q2 = -3Q1. How is the charge distributed on the sphere? Q2 Q1 R1 R2 (A) There is no charge on the sphere. (B) The charge is uniformly distributed on the outside surface of the sphere. 1/26/2016 (C) The charge is uniformly distributed throughout the sphere. 17 Question 2. • A solid conducting sphere is concentric with a thin conducting shell, as shown. • The inner sphere carries a charge Q1, and the spherical shell carries a charge Q2, such that Q2 = -3Q1. How is the charge distributed on the spherical shell? Q2 Q1 R1 R2 (A) There is no charge on the shell. (B) The charge is uniformly distributed on the outside surface of the shell. (C) The charge is uniformly distributed on the inner and outer surfaces of the shell. 1/26/2016 19 Question 3. • A solid conducting sphere is concentric with a thin conducting shell, as shown. • The inner sphere carries a charge Q1, and the spherical shell carries a charge Q2, such that Q2 = -3Q1. Q2 Q1 R1 What is the electric field at r < R1? A) E 0 R2 1 Q1 E B) 4 0 r12 C) 1/26/2016 1 -3Q E 4 0 r 2 21 Question 4. • A solid conducting sphere is concentric with a thin conducting shell, as shown. • The inner sphere carries a charge Q1, and the spherical shell carries a charge Q2, such that Q2 = -3Q1. Q2 Q1 R1 What is the electric field at R1<r < R2? A) E 0 R2 1 Q1 E B) 4 0 r12 C) 1/26/2016 1 -3Q E 4 0 r 2 23 Question 5. • A solid conducting sphere is concentric with a thin conducting shell, as shown. • The inner sphere carries a charge Q1, and the spherical shell carries a charge Q2, such that Q2 = -3Q1. Q2 Q1 R1 What is the electric field at R2<r A) 1 Q1 E 4 0 r12 B) 1 Q1 + Q2 E 4 0 r12 C) 1/26/2016 1 -3Q1 E 4 0 r 2 R2 24 Question 6 • A solid conducting sphere is concentric with a thin conducting shell, as shown. • The inner sphere carries a charge Q1, and the spherical shell carries a charge Q2, such that Q2 = -3Q1. Q2 Q1 R1 R2 What happens when you connect the two spheres with a wire? (A) The charge is uniformly distributed on the outside surface of the shell. (B) There is no charge on the sphere or the shell. (C) The charge is uniformly distributed on the outer surfaces of the sphere and the shell. 1/26/2016 25 Example Consider the following two topologies: A) B) A solid non-conducting sphere carries a total charge Q = -3 C distributed evenly throughout. It is surrounded by an uncharged conducting spherical shell. s2 s1 -Q E Same as (A) but conducting shell removed •Compare the electric field at point X in cases A and B: (a) EA < EB (b) EA = EB (c) EA > EB •Select a sphere passing through the point X as the Gaussian surface. •How much charge does it enclose? •Answer: -Q, whether or not the uncharged shell is present. (The field at point X is determined only by the objects with NET CHARGE.) 1/26/2016 28 Conductors: External Electric Field 1/26/2016 33 Two Parallel Conducting Sheets Find the electric field to the left of the sheets, between the sheets and to the right of the sheets. 1/26/2016 34 Uniform Charge Density: Summary Non-conductor Cylindrical symmetry Planar Spherical symmetry r E 2 0 R2 E 2 0 r s E 2 0 1 Q E r 3 4 0 R 1 1/26/2016 Q E 2 4 0 r Conductor E 0 inside E 2 0 r s E 0 outside inside E 0 1 Q E 2 4 0 r outside 35 Summary of Lectures 3, 4 & 5 *Relates net flux, F, of an electric field through a closed surface to the net charge that is enclosed by the surface. o F o E dA qenc *Takes advantage of certain symmetries (spherical, cylindrical, planar) *Gauss’ Law proves that electric fields vanish in conductor extra charges reside on surface 1/26/2016 36