4Chapter 4Electric FluxGauss’s LawExample 4.1: Infinitely Long Rod of Uniform Charge DensityExample 4.2: Infinite Plane of ChargeExample 4.3: Spherical ShellExample 4.4: Non-Conducting Solid SphereConductorsExample 4.5: Conductor with Charge Inside a CavityExample 4.6: Electric Potential Due to a Spherical ShellForce on a ConductorSummaryAppendix: Tensions and PressuresAnimation 4.1: Charged Particle Moving in a Constant ElectrAnimation 4.2: Charged Particle at Rest in a Time-Varying FiAnimation 4.3: Like and Unlike Charges Hanging from PendulumProblem-Solving StrategiesSolved ProblemsTwo Parallel Infinite Non-Conducting PlanesElectric Flux Through a Square SurfaceGauss’s Law for GravityElectric Potential of a Uniformly Charged SphereConceptual QuestionsAdditional ProblemsNon-Conducting Solid Sphere with a CavityP-N JunctionSphere with Non-Uniform Charge DistributionThin SlabElectric Potential Energy of a Solid SphereCalculating Electric Field from Electrical PotentialChapter 4 Gauss’s Law 4.1 Electric Flux..........................................................................................................4-2 4.2 Gauss’s Law..........................................................................................................4-3 Example 4.1: Infinitely Long Rod of Uniform Charge Density.............................4-8 Example 4.2: Infinite Plane of Charge.................................................................... 4-9 Example 4.3: Spherical Shell................................................................................4-12 Example 4.4: Non-Conducting Solid Sphere........................................................4-13 4.3 Conductors..........................................................................................................4-15 Example 4.5: Conductor with Charge Inside a Cavity .........................................4-18 Example 4.6: Electric Potential Due to a Spherical Shell..................................... 4-19 4.4 Force on a Conductor..........................................................................................4-22 4.5 Summary.............................................................................................................4-23 4.6 Appendix: Tensions and Pressures .....................................................................4-24 Animation 4.1: Charged Particle Moving in a Constant Electric Field...............4-25 Animation 4.2: Charged Particle at Rest in a Time-Varying Field .....................4-27 Animation 4.3: Like and Unlike Charges Hanging from Pendulums..................4-28 4.7 Problem-Solving Strategies ................................................................................4-29 4.8 Solved Problems .................................................................................................4-31 4.8.1 Two Parallel Infinite Non-Conducting Planes.............................................4-31 4.8.2 Electric Flux Through a Square Surface...................................................... 4-32 4.8.3 Gauss’s Law for Gravity..............................................................................4-34 4.8.4 Electric Potential of a Uniformly Charged Sphere ......................................4-34 4.9 Conceptual Questions .........................................................................................4-36 4.10 Additional Problems .........................................................................................4-36 4.10.1 Non-Conducting Solid Sphere with a Cavity.............................................4-36 4.10.2 P-N Junction...............................................................................................4-36 4.10.3 Sphere with Non-Uniform Charge Distribution ........................................4-37 4.10.4 Thin Slab....................................................................................................4-37 4.10.5 Electric Potential Energy of a Solid Sphere...............................................4-38 4.10.6 Calculating Electric Field from Electrical Potential .................................. 4-38 4-1Gauss’s Law 4.1 Electric Flux In Chapter 2 we showed that the strength of an electric field is proportional to the number of field lines per area. The number of electric field lines that penetrates a given surface is called an “electric flux,” which we denote as EΦ. The electric field can therefore be thought of as the number of lines per unit area. Figure 4.1.1 Electric field lines passing through a surface of area A. Consider the surface shown in Figure 4.1.1. Let ˆA=Anr be defined as the area vector having a magnitude of the area of the surface, , and pointing in the normal direction, . If the surface is placed in a uniform electric field EAˆnur that points in the same direction as , i.e., perpendicular to the surface A, the flux through the surface is ˆn ˆEAEAΦ= ⋅ = ⋅ =EA Enrrr (4.1.1) On the other hand, if the electric fieldEur makes an angle θ with (Figure 4.1.2), the electric flux becomes ˆn ncosEEA E AθΦ= ⋅ = =EArr (4.1.2) where is the component of EnˆE =⋅Enrrperpendicular to the surface. Figure 4.1.2 Electric field lines passing through a surface of area A whose normal makes an angle θ with the field. 4-2Note that with the definition for the normal vector , the electric flux is positive if the electric field lines are leaving the surface, and negative if entering the surface. ˆnEΦ In general, a surface S can be curved and the electric field Eur may vary over the surface. We shall be interested in the case where the surface is closed. A closed surface is a surface which completely encloses a volume. In order to compute the electric flux, we divide the surface into a large number of infinitesimal area elements , as shown in Figure 4.1.3. Note that for a closed surface the unit vector is chosen to point in the outward normal direction. ˆiiA∆=∆Anriˆin Figure 4.1.3 Electric field passing through an area element i∆Ar, making an angle θ with the normal of the surface. The electric flux through is i∆Ar cosEi iiiEAθ∆Φ = ⋅∆ = ∆EArr (4.1.3) The total flux through the entire surface can be obtained by summing over all the area elements. Taking the limit and the number of elements to infinity, we have 0i∆→Ar 0limEiiiSAd∆→Φ= ⋅ = ⋅d∑∫∫EA EArrrrÒ (4.1.4) where the symboldenotes a
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