Question 3 A solid conducting sphere is concentric

Question 3. • A solid conducting sphere is concentric with a thin conducting shell, as shown. • The inner sphere carries a charge Q 1, and the spherical shell carries a charge Q 2, such that Q 2 = -3 Q 1. What is the electric field at r < R 1? Q 2 Q 1 R 2 A) B) C) 10/30/2020 1

Question 4. • A solid conducting sphere is concentric with a thin conducting shell, as shown. • The inner sphere carries a charge Q 1, and the spherical shell carries a charge Q 2, such that Q 2 = -3 Q 1. What is the electric field at R 1<r < R 2? Q 2 Q 1 R 2 A) B) C) 10/30/2020 3

Question 5. • A solid conducting sphere is concentric with a thin conducting shell, as shown. • The inner sphere carries a charge Q 1, and the spherical shell carries a charge Q 2, such that Q 2 = -3 Q 1. What is the electric field at R 2<r Q 2 Q 1 R 2 A) B) C) 10/30/2020 4

Question 6 • A solid conducting sphere is concentric with a thin conducting shell, as shown. • The inner sphere carries a charge Q 1, and the spherical shell carries a charge Q 2, such that Q 2 = -3 Q 1. Q 2 Q 1 R 2 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. 10/30/2020 5

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. s 2 s 1 -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. ) 10/30/2020 8

Conductors: External Electric Field 10/30/2020 9

Two Parallel Conducting Sheets Find the electric field to the left of the sheets, between the sheets and to the right of the sheets. 10/30/2020 10

Uniform Charge Density: Summary Non-conductor Cylindrical symmetry Conductor inside outside Planar Spherical symmetry inside outside 10/30/2020 11

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. *Takes advantage of certain symmetries (spherical, cylindrical, planar) *Gauss’ Law proves that electric fields vanish in conductor, extra charges reside on surface 10/30/2020 12

Lectures 6 & 7: Chapter 23 Electric Potential Q V 4 pe 0 R Q 4 pe 0 r R Definitions R C r Examples R B r B q r A A Path independence 10/30/2020 Equipotential surfaces 13

From Mechanics (PHYS 172) • Energy Kinetic Energy: associated with the state of motion Potential Energy: associated with the configuration of the system • Conservative Forces: Work done by a conservative force is independent of path 10/30/2020 14

From Mechanics (PHYS 172) • Work F dr or F dr 10/30/2020 W>0 Object speeds up ( K > 0 ) W<0 Object slows down ( K < 0 ) W=0 Constant speed ( K = 0 ) 15

Electric Potential Energy When an electrostatic force acts between two or more charges within a system, we can assign an Electric Potential Energy: + + + 10/30/2020 + F x If a Coulomb force does negative work Potential energy increases 16

Question You hold a positively charged ball and walk due west in a region that contains an electric field directed due east. + West E East WH is the work done by the hand on the ball WE is the work done by the electric field on the ball Which of the following statements is true: A) B) C) D) 10/30/2020 WH > 0 WH < 0 and WE > 0 and WE < 0 and WE > 0 17

Example: Electric Potential Energy What is the change in electrical potential energy of a released electron in the atmosphere when the electrostatic force from the near Earth’s electric field (directed downward) causes the electron to move vertically upwards through a distance d? 1. U of the electron is related to the work done on it by the electric field: 2. Work done by a constant force on a particle undergoing displacement: 3. Electrostatic Force and Electric Field are related: 10/30/2020 18

Example: Electric Potential Energy What is the change in electrical potential energy of a released electron in the atmosphere when the electrostatic force from the near Earth’s electric field (directed downward) causes the electron to move vertically upwards through a distance d? 1. U of the electron is related to the work done on it by the electric field: Key Idea: 2. Work done by a constant force on a particle undergoing displacement: Key Idea: 3. Electrostatic Force and Electric Field are related: Key Idea: 10/30/2020 Electric potential decreases as electron rises. 19

Electric Potential versus Electrical Potential Energy Electric Potential is a property of an electric field and is measured in J/C or V Electric Potential Energy is an energy of system consisting of the charged object and the external electric field, and is measured in Joules. 10/30/2020 23

Potential & Electric Fields The electric field points in the direction in which the potential decreases most rapidly. 10/30/2020 24

Example: Potential Difference *independent of path i (a) What is V moving directly from point i to point f? c f (b) What is V moving from point i to point c to point f? 10/30/2020 26

Question: • A single charge ( Q = -1 C) is fixed at the origin. Define point A at x = + 5 m and point B at x = +2 m. – What is the sign of the potential difference between A and B? – Where VBA = VB-VA (a) VBA < 0 10/30/2020 (b) VBA = 0 -Q B ´ -1 C A ´ x (c) VBA > 0 27

Potential due to a point charge: Find V in space around a charged particle relative to the zero potential at infinity: + 10/30/2020 28

V(r) versus r for a positive charge at r = 0 V(r) to r For a point charge 10/30/2020 29

Electrical Potential Energy Push q 0 “uphill” and its electrical potential energy increases according to 10/30/2020 30

Demo 5 A-16 + + + R + + + + + V(r) + Gauss’ law says the sphere looks like a point charge outside R. 10/30/2020 31

Demo 5 A-35 Get energy out charge flow + + + R + + + + Also try an elongated neon bulb. 10/30/2020 r 1 r 2 across fluorescent light bulb 32

Potential due to a Group of Point Charges Find the Potential at the center of the square. d = 1. 3 m q 1 = +12 n. C + + q 3 =+31 n. C 10/30/2020 q 2 = -24 n. C - + q 3 =+17 n. C 33

Electrical Potential Energy of a System of Point Charges U of a system of fixed point charges equals W done by an external agent to assemble the system by bringing each charge in from infinity. + + If q 1 & q 2 have the same sign, we must do positive work to push against mutual repulsion. If q 1 & q 2 have opposite signs, we must do negative work against mutual attraction 10/30/2020 35

Calculating the Electric Field from the Potential Field 10/30/2020 36

Potential Energy of an Electric Dipole Potential energy can be associated with the orientation of an electric dipole in an electric field. U is least =0 U=-p. E U is greatest =180 U=p. E U =0 when =90