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. R1 R2 Q1 Q2 What is the electric field at r < R1? A) B) C) 11/27/2018

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. R1 R2 Q1 Q2 What is the electric field at R1<r < R2? A) B) C) 11/27/2018

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. R1 R2 Q1 Q2 What is the electric field at R2<r A) B) C) 11/27/2018

Question 6 R1 R2 Q1 Q2 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. 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. 11/27/2018

Example E Compare the electric field at point X in cases A and B: Consider the following two topologies: A) 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 B) 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.) 11/27/2018

Conductors: External Electric Field For a spherical conductor, excess charge distributes itself uniformly For a non-spherical conductor, the surface density varies over the surface & makes the E field difficult to determine. However, the E field set-up just outside the conductor is easy to determine. Examine a tiny portion of a large conductor with an excess of positive charge. 11/27/2018

Two Parallel Conducting Sheets Find the electric field to the left of the sheets, between the sheets and to the right of the sheets. 11/27/2018

Uniform Charge Density: Summary Cylindrical symmetry Planar Spherical Non-conductor Conductor inside outside 11/27/2018

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 11/27/2018

Lectures 6 & 7: Chapter 23 Electric Potential Definitions Examples C B V Q 4pe0 r 4pe0 R Definitions Examples C B r B A r q A Equipotential surfaces Path independence 11/27/2018

From Mechanics (PHYS 172) Energy Conservative Forces: Kinetic Energy: associated with the state of motion Potential Energy: associated with the configuration of the system Work done by a conservative force is independent of path 11/27/2018

From Mechanics (PHYS 172) Work F W > 0 dr W < 0 F dr F W = 0 dr Object speeds up ( DK > 0 ) W < 0 Object slows down (DK < 0 ) F dr or F dr W = 0 Constant speed (DK = 0 ) 11/27/2018

Electric Potential Energy When an electrostatic force acts between two or more charges within a system, we can assign an Electric Potential Energy: F + + Dx + + If a Coulomb force does negative work Potential energy increases 11/27/2018

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) WH > 0 and WE > 0 B) WH > 0 and WE < 0 C) WH < 0 and WE < 0 D) WH < 0 and WE > 0 Work by hand is positive Work by electric field is negative 11/27/2018

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? U of the electron is related to the work done on it by the electric field: Work done by a constant force on a particle undergoing displacement: Electrostatic Force and Electric Field are related: 11/27/2018

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? U of the electron is related to the work done on it by the electric field: Work done by a constant force on a particle undergoing displacement: Electrostatic Force and Electric Field are related: Key Idea: Key Idea: Key Idea: Electric potential decreases as electron rises. 11/27/2018

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. 11/27/2018

Potential & Electric Fields The electric field points in the direction in which the potential decreases most rapidly. 11/27/2018

Example: Potential Difference *independent of path (a) What is V moving directly from point i to point f? i c f (b) What is V moving from point i to point c to point f? 11/27/2018

Question: x ´ -Q (a) VBA < 0 (b) VBA = 0 (c) VBA > 0 A single charge ( Q = -1C) is fixed at the origin. Define point A at x = + 5m and point B at x = +2m. What is the sign of the potential difference between A and B? Where VBA = VB-VA x -1C ´ A B -Q (a) VBA < 0 (b) VBA = 0 (c) VBA > 0 11/27/2018

Potential due to a point charge: Find V in space around a charged particle relative to the zero potential at infinity: + 11/27/2018

V(r) versus r for a positive charge at r = 0 to r For a point charge 11/27/2018

Electrical Potential Energy Push q0 “uphill” and its electrical potential energy increases according to 11/27/2018

+ Demo 5A-16 R + + V(r) + R + + + + + + + Gauss’ law says the sphere looks like a point charge outside R. 11/27/2018

Demo 5A-35 + + + R + + r1 r2 + + + + + + across fluorescent light bulb Get energy out charge flow + + + R + + r1 r2 + + + + + + across fluorescent light bulb Also try an elongated neon bulb. 11/27/2018

Potential due to a Group of Point Charges Find the Potential at the center of the square. q1 = +12 nC q2 = -24 nC + - d = 1.3 m q3 =+31 nC q3 =+17 nC 11/27/2018

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 q1 & q2 have the same sign, we must do positive work to push against mutual repulsion. If q1 & q2 have opposite signs, we must do negative work against mutual attraction 11/27/2018

Calculating the Electric Field from the Potential Field 11/27/2018

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=-pE U is greatest =180 U=pE U =0 when =90