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Chapter 22 Gauss’s Law PowerPoint® Lectures for University Physics, Thirteenth Edition – Hugh D. Young and Roger A. Freedman Lectures by Wayne Anderson Copyright © 2012 Pearson Education Inc. Goals for Chapter 22 • To use the electric field at a surface to determine the charge within the surface • To learn the meaning of electric flux and how to calculate it • To learn the relationship between the electric flux through a surface and the charge within the surface • To use Gauss’s law to calculate electric fields • To learn where the charge on a conductor is located Copyright © 2012 Pearson Education Inc. Introduction • Would the girl’s hair stand on end if she were inside the metal sphere? • Symmetry properties play an important role in physics. • Gauss’s law will allow us to do electric-field calculations using symmetry principles. Copyright © 2012 Pearson Education Inc. Charge and electric flux • Positive charge within the box produces outward electric flux through the surface of the box, and negative charge produces inward flux. (See Figure 22.2 below.) Copyright © 2012 Pearson Education Inc. Figure 22.3 Zero net charge inside a box • There are three ways in which there is zero net charge inside a box and no net electric flux through the surface of the box. Copyright © 2012 Pearson Education Inc. What affects the flux through a box? • As Figure 22.4 below shows, doubling the charge within the box doubles the flux, but doubling the size of the box does not change the flux. Why? Copyright © 2012 Pearson Education Inc. Calculating electric flux • The electric flux depends on the intensity of the electric field E and the orientation of the surface with respect to the electric field. Copyright © 2012 Pearson Education Inc. Example 22.1 Electric flux through a disk Copyright © 2012 Pearson Education Inc. Example 22.2 Electric flux through a cube An imaginary cube of side L is in a region of uniform electric field E. Find the electric flux through each surface of the cube when (a) it is oriented with two of its faces perpendicular to E. Insanity is doing the same thing over and over and expecting a different result. Albert Einstein Copyright © 2012 Pearson Education Inc. Gauss’s Law • Gauss’s law is an alternative to Coulomb’s law and is completely equivalent to it. • What did you find out about Gauss? 1. 2. 3. 4. 5. Copyright © 2012 Pearson Education Inc. Figure 22.11 Point charge centered in a spherical surface • The flux through the sphere is independent of the size of the sphere and depends only on the charge inside. Copyright © 2012 Pearson Education Inc. Point charge inside a nonspherical surface • As before, the flux is independent of the surface and depends only on the charge inside. (See Figure 22.12 below.) Copyright © 2012 Pearson Education Inc. Example 22.3 Electric flux through a sphere A point charge q=+3.0 µC is surrounded by an imaginary sphere of radius = 0.20 m centered on the charge. Find the resulting electric flux through the sphere. Copyright © 2012 Pearson Education Inc. Positive and negative flux • Flux is positive (outward) if the enclosed charge is positive, and negative (inward) if the charge is negative. Copyright © 2012 Pearson Education Inc. Example 22.4 Electric flux and enclosed charge • What is the electric flux through each of the following surfaces: A, B, C and D. Copyright © 2012 Pearson Education Inc. Gaussian Surfaces Gaussian surfaces are imaginary. We use them as a convenience, to ease the calculation of electric flux, electric field, area or enclosed charge. What Gaussian surface did you bring with you tonight? While any closed surface could be a Gaussian surface, we typically use just a few specific types. What are they? Why are they so useful? Copyright © 2012 Pearson Education Inc. Applications of Gauss’s law • Under electrostatic conditions, any excess charge on a conductor resides entirely on its surface. Our EE’s have a name for this, what is it? • Example 22.5 uses Figure 22.18 below right. This answers the question on slide 3 Copyright © 2012 Pearson Education Inc. Example 22.6 Field of a line charge Electric charge is distributed uniformly along an infinitely long, thin wire. The charge per unit length is λ (assumed positive). Find the electric field using Gauss’s Law. Looking from the end of the wire. The electric field radiates out radially from the wire. Therefore only the circumference of the cylinder contributes to the electric flux. Copyright © 2012 Pearson Education Inc. Notice that E is inverse r, not r2, in this case. Example 22.7 Field of an infinite sheet of charge Use Gauss’s Law to find the electric field caused by a thin, flat, infinite sheet with a uniform positive surface charge density σ. In this case, the electric flux emerges only from the ends. Could we have used a different Gaussian surface to solve this problem? If so, which one? Copyright © 2012 Pearson Education Inc. Example 22.8 Field between two oppositely charged parallel conducting plates But if the plates are large compared to the distance between them…….. Copyright © 2012 Pearson Education Inc. Example 22.9 A uniformly charged sphere We already know the electric field outside the sphere. What is the electric field inside a uniformly charged sphere? Copyright © 2012 Pearson Education Inc. Charges on conductors Copyright © 2012 Pearson Education Inc. Conceptual Example 22.11 A conductor with a cavity A solid conductor with a cavity carries a total charge of + 7 nC. Within the cavity, insulated from the conductor is a point charge of -5 nC. How much charge is on each surface (inner and outer) of the conductor? Copyright © 2012 Pearson Education Inc. Testing Gauss’s law experimentally • Faraday’s Ice Pail Experiment http://www.youtube.com/watch?v=GNizWxAD-9M Copyright © 2012 Pearson Education Inc. The Van de Graaff generator Note that there is a flow of electrons. That is where we are going next in our study of electricity. Copyright © 2012 Pearson Education Inc. Electrostatic shielding • A conducting box (a Faraday cage) in an electric field shields the interior from the field. • Note the construction of the cage. Copyright © 2012 Pearson Education Inc. Electric field near the surface of a conductor • The electric field at the surface of any conductor is always perpendicular to the surface and has magnitude E=/0. We can apply this equation as long as we are close to the surface, compared to the size of the conductor. Copyright © 2012 Pearson Education Inc. Example 22.13 Electric field of the Earth The Earth has a net electric charge. The resulting electric field near the surface of the Earth has an average value of about 150 N/C, directed toward the center of the Earth (a) What is the Earth’s charge density? (b) What is the total surface charge of the Earth? Copyright © 2012 Pearson Education Inc. Goals for Chapter 22 • To use the electric field at a surface to determine the charge within the surface • To learn the meaning of electric flux and how to calculate it • To learn the relationship between the electric flux through a surface and the charge within the surface • To use Gauss’s law to calculate electric fields • To learn where the charge on a conductor is located Copyright © 2012 Pearson Education Inc.