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Chapter 22 -Gauss’s Law I. Introduction Another method of determining the values of electric fields. Also a guide to understanding how charges will be distributed II. Define the Electric Flux, , through an area A. Qualitatively: is a measure of the number of lines of force passing through the area On what should the electric flux, , depend? B. 1. Strength of the electric field – remember that the density of the number of lines of force is proportional to the strength of the electric field, that is, E 2. Size of the area – the larger the area, the more lines of force passing through it, A 3. Orientation of the electric field to the area 4. Digression: the area as a vector Quantitatively: 1. Uniform electric field, E = constant area, A A E A cos perspective view side view 22-1 area, A E Define the electric flux through and area A placed in a uniform electric field E as: = EA cos = E A , Nm2 where the direction of A is perpendicular to the surface. The units are: C Example: uniform electric field A uniform electric field, E 2000iˆ 3000kˆ N/C, is present in space. Find the electric flux through each of the surfaces of the triangular block shown. y 4m B C 3m A E x F 2m 5m D z 2. Nonuniform electric field, E varies Look at the electric flux due to a nonuniform electric field passing through a small area A The electric field is relatively constant through the small area A , so the flux passing through the area can be written as an approximation: E A E A 22-2 Example: nonuniform electric field An electric field, E 150yiˆ N/C, exists in space. Find the flux through each of the faces of the cube whose sides are 2 meters long. y x z III. Electric Flux through a Gaussian Surface (a closed surface) A. Gaussian or closed surface: B. Signs of the electric flux, Gaussian or closed surface Note the signs of the electric flux in the examples, above. 22-3 C. Find the value of the flux through a closed surface. Remember: E E dA 1. Qualitatively: a. no charges inside the closed surface b. a charge +Q inside the closed surface c. a charge –Q inside the closed surface 22-4 d. 2. net charge is zero inside the closed surface Quantitatively: a. Note the amount of charge inside the closed surfaces in examples 3 and 4. b. A point charge +q inside the closed surface. dA +q r dA E d r d dA' dA’ dA' r 2 d 22-5 dA E 3. Gauss’s Law Q E E dA encl o IV. Applications of Gauss’s Law Remember that Gauss’s Law can be used qualitatively to determine the location of charges and quantitatively to determine the strength of the electric field for relatively simple charge distributions. Remember also that in electrostatics, E = 0 inside a conductor. (Do you remember why?) A. Location of excess charge 1. Excess charge, -4Q, placed on a solid conductor 2. A hollow conductor with a charge placed inside the hollow part. +3Q 3. A hollow conductor with a charge –2Q placed inside the hollow part and charge –5Q placed on the conductor -5Q -2Q 22-6 B. Quantitative calculation of the electric field using Gauss’s law Construct a closed surface (Gaussian surface) in the region where you want to find the strength of the electric field such that: - surfaces are perpendicular to the electric field, and the electric field has a constant value on the surfaces (use ideas of symmetry to argue direction and magnitude of the electric field and note then that the electric field E can be extracted from the integral E dA . - and, if necessary to close the surfaces, choose other surfaces that are parallel to the electric field so that E dA is zero 1. E-field of a point charge, +Q. E +Q r 2. Excess charge Q placed on a spherical conductor of radius R. Find the E-field both inside and outside the sphere. R E r 22-7 3. Charge Q uniformly distributed throughout the volume of a sphere of radius R. (Is the sphere a conductor?) Find the E-field both inside and outside the sphere. R E r A sphere of radius R has a charge density = Ar, where A is a constant and r is the radial distance from the center of the sphere. Field the E-field both inside and outside the sphere. 4. R E r 22-8 5. Charges uniformly distributed on a long, thin wire. The charge density is Find the electric at a perpendicular distance r from the wire. 6. A coaxial cable with linear charge density on the inner conductor and on the outer conductor. The radius of the inner conductor is a, the inside radius of the outer conductor is b, and the outside radius is c. Find the E-field in all four regions. b c a end on view 22-9 7. A large, flat surface with a single layer of charge uniformly distributed over the surface. The surface charge density is Find the E-field a distance z above the surface. 8. Charge placed on a large, flat conductor. Find the E-field above, below and inside the conductor. 9. Charge is placed on an arbitrarily shaped conductor. Find the E-field close to the surface on the conductor. 22-10