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.) 9/13/2018

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. 9/13/2018

Two Parallel Conducting Sheets Find the electric field to the left of the sheets, between the sheets and to the right of the sheets. 9/13/2018

Uniform Charge Density: Summary Cylindrical symmetry Planar Spherical Non-conductor Conductor inside outside 9/13/2018

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 9/13/2018

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 9/13/2018

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 9/13/2018

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 ) 9/13/2018

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 9/13/2018

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: 9/13/2018

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. 9/13/2018

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. 9/13/2018

Potential & Electric Fields The electric field points in the direction in which the potential decreases most rapidly. 9/13/2018

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? 9/13/2018

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 9/13/2018

Potential due to a point charge: Find V in space around a charged particle relative to the zero potential at infinity: + 9/13/2018

V(r) versus r for a positive charge at r = 0 to r For a point charge 9/13/2018

Electrical Potential Energy Push q0 “uphill” and its electrical potential energy increases according to 9/13/2018

+ Demos: R + + V(r) + R + + + + + + + Gauss’ law says the sphere looks like a point charge outside R. 9/13/2018

Demo + + + 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. 9/13/2018

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 9/13/2018

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 9/13/2018

Calculating the Electric Field from the Potential Field 9/13/2018

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