Gauss’ Law Symmetry ALWAYS TRUE! In cases with symmetry can pull E outside and get Spherical Cylindrical Planar 11/13/2018

Planar Symmetry: Infinite Sheet of Charge Non-Conducting Sheet + + + + + + + + s+ + + + + 11/13/2018

Example: Two Parallel Non-Conducting Sheets 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. ++++++ - 11/13/2018

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. Properties of conductors & insulators are due to the structure & electrical nature of atoms. Atoms consist of protons, neutrons & electrons. Protons and neutrons are packed tightly in the nucrleus and the electrons orbit the nucleus. Charge of electron and proton have the same magnitude but different signs. Neutral atoms contain equal amounts of protons and electrons. The electrons are held near the nucleus because they have the opposite electrical sign of the protons & they are attracted to the nucleus. Electrons are bound to their atoms & cannot move from place to place. Conductors: metals, tap water, human body…..Atoms in a conductor allow one or more of their outermost electrons to become detached. These conduction electrons can move freely throughout the conductor…leaving behind positively charged atoms (positive ions). Insulators: Air, glass , plastic Semi-conductors: silicon, germanium When charge moves an electric current exists. Most materials (even good conductors) tend to resist flow of charge. Super-conductors: Resistance is zero. 11/13/2018

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. We can use Gauss’ law to prove the above theorem about isolated conductors. The theorem seems reasonable since charges with the same sign repel each other….by moving to the surface. The added charges are getting as far away from each other as they can. Let’s take a non-symmetric isolated lump of copper hanging from an insulating thread and having excess charge. If E internal was not equal to zero, the field would exert forces on the free conduction electrons & thus current would always exist within a conductor. OF course, no such perpetual currents exist in an isolated conductor. So the internal E field IS equal to zero. As the object is charged, there is an internal field, however, the charge redistributtes quickly so Enet internal equal zero. The movement of charge ceases because the net force on each charge is equal to zero…..Electrostatic Equilibrium 11/13/2018

Conductors Isolator Conductor with a Cavity Cavity is totally within the conductor. Because E=0 inside the conductor, the walls of the cavity have no net charge. All excess charge remains on the surface of the conductor. 11/13/2018

Gauss’ Law: Spherical Symmetry Demo: 5A-13 No Internal Field Electric Field inside and outside a shell of uniform charge distribution 11/13/2018

Conductors: Shielding from an Electrical Field Electric field lines for an oppositely charged metal cylinder and metal plate. Note that: Electric field lines are perpendicular to the conductors. There are no electric field lines inside the cylinder. 11/13/2018

Conductors: Shielding from an Electrical Field student sensor sparks screened cage Van de Graaff generator 11/13/2018

Conductors DEMO: 5A-12 Charge within a Conductor Faraday’s Pail: shows that all charge moves to the outside of the pail. Charge wand, touch inside of bucket, remove wand test on another electroscope to show the charge on it is zero. 5A-12: Hollow sphere: charge the sphere. Touch wand to inside & then bring to electroscope to show zero charge. Then touch outside & the check on electroscope…it will have a charge. 11/13/2018

Conductors Charge Distribution on a Conductor DEMO: 5A-21 Charge Distribution on a Conductor http://cnx.org/content/m42317/latest/Figure_19_07_07a.jpg So far we have considered excess charges on a smooth, symmetrical conductor surface. What happens if a conductor has sharp corners or is pointed? Excess charges on a nonuniform conductor become concentrated at the sharpest points. Additionally, excess charge may move on or off the conductor at the sharpest points.To see how and why this happens, consider the charged conductor in Figure 6. The electrostatic repulsion of like charges is most effective in moving them apart on the flattest surface, and so they become least concentrated there. This is because the forces between identical pairs of charges at either end of the conductor are identical, but the components of the forces parallel to the surfaces are different. The component parallel to the surface is greatest on the flattest surface and, hence, more effective in moving the charge.The same effect is produced on a conductor by an externally applied electric field, as seen in Figure 6 (c). Since the field lines must be perpendicular to the surface, more of them are concentrated on the most curved parts. FIGURE 6: Excess charge on a nonuniform conductor becomes most concentrated at the location of greatest curvature. (a) The forces between identical pairs of charges at either end of the conductor are identical, but the components of the forces parallel to the surface are different. It isF∥that moves the charges apart once they have reached the surface. (b)F∥is smallest at the more pointed end, the charges are left closer together, producing the electric field shown. (c) An uncharged conductor in an originally uniform electric field is polarized, with the most concentrated charge at its most pointed end. http://cnx.org/content/m42317/latest/Figure_19_07_07a.jpg 11/13/2018

Example: -50e -100e A ball of charge -50e lies at the center of a hollow spherical metal shell that has a net charge -100e. (1) What is the charge on the shell’s inner surface? (a) -50e (b) 0 (c ) +50e 11/13/2018

Example: -50e -100e A ball of charge -50e lies at the center of a hollow spherical metal shell that has a net charge -100e. (2) What is the charge on the shell’s outer surface? (a) -150e (b) -50e (c) +100e 11/13/2018

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. R1 R2 Q1 Q2 How is the charge distributed on the sphere? There is no charge on the sphere. (B) The charge is uniformly distributed on the outside surface of the sphere. (C) The charge is uniformly distributed throughout the sphere. 11/13/2018

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. R1 R2 Q1 Q2 How is the charge distributed on the spherical shell? 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. 11/13/2018

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) 11/13/2018

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) 11/13/2018

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) 11/13/2018

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. 11/13/2018

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.) 11/13/2018

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. 11/13/2018

Two Parallel Conducting Sheets Find the electric field to the left of the sheets, between the sheets and to the right of the sheets. 11/13/2018

Uniform Charge Density: Summary Cylindrical symmetry Planar Spherical Non-conductor Conductor inside outside 11/13/2018

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 11/13/2018