Gauss’s Law and Symmetry

1. Gauss’s Law and Symmetry Kenny Smart Ethan Hatke

2. Electric Flux • Flux is the amount of an electric field that goes through an area • Flux of the electric field is a scalar quantity

3. Gauss’s Law • Gauss’s Law relates the total electric flux through a closed surface, the total charge enclosed by the surface, and the electric permittivity of free space • Gauss’s Law is commonly used to find the strength of an electric field produced by a single charge or a continuous distribution of charges • The area enclosing a charge we call a Gaussian shape, and is not a real object, just a tool to help solve electrostatic problems • Electric flux is negative when it is into the Gaussian surface, and positive when it is out of the Gaussian surface

4. λ,σ,ρ • λ is the linear charge density, usually of a wire, and describes the amount of charge per unit length • λ*L • σ is the area charge density, usually of a sheet or plate of charge, and describes the amount of charge per unit area • σ *A • ρ is the volumetric charge density, usually of nonconductors, and describes the amount of charge per unit volume • ρ*V

5. Conducting v. Nonconducting • Conductors can’t hold an electric field inside their surface, all the like-charges repel each other and move to the outer radius of the object • Nonconductors can hold an electric field inside their surface because the charge is not free to move throughout the object and there are charges within the object, usually is a uniform field throughout the object but doesn’t have to be

6. Spherical Symmetry • Within the inner sphere: No charge is enclosed so, =0 and there is no electric field within • In the donut at any point up to the outer radius, there is no net charge enclosed, and so there is no electric field and =0 • At the outer radius of the donut: the charge enclosed is +Q and radius of R:, so (this is equivalent to Coulomb’s Law) • Beyond the outer radius, the electric field will decrease at a rate of 1/r^2 because the Gaussian shape is a sphere Conducting sphere with charge +Q, inner radius r and outer radius of R

7. Cylindrical Symmetry with a line of charge Wire has linear charge density, λ, length L and radius r • A wire of charge is enclosed by a Gaussian cylinder • Qenclosed=λ*L • Area=2πrL