Gauss' Law

Enclosing the earth in a sphere of a radius equal to the radius of the earth plus the height from the surface, we can solve for gravitational acceleration. *Note - our value for G should actually be 6.67*10^-11. 

A cylindrical gaussian surface located inside a long conducting wire shares the same linear charge density as the wire. We therefore can create a relationship between the charge q inside the smaller cylinder of radius r and the charge Q inside the wire of radius R. Solving for the flux through the gaussian surface, we can replace q with the relationship formed between the two charges.

Inducing a charge on the surface of a conducting cylinder will cause any excess of charge to lie on the surface. Therefore, if we were to place pieces of foil paper on the cylinder, they would eventually experience a charge as well. As this occurs, the charge on the foil paper repels away from the charge on the cylinder.

This Van de Graff  electrostatic generator is what was used to induce the charge on the cylinder.

A sphere of radius r inside a larger sphere of radius r shares the same charge density (Q/V). This allows us to create a relationship between the charge inside both spheres and the radius of the spheres (Q/R^3 = q/r^3). Since the flux through the sphere smaller radius is E dot 4*pi*r^2, and the charge enclosed by that sphere is Q*r^3/R^3, we see that we can solve for the electric field caused the charge enclosed in the smaller sphere. Also, we can see that if the radius of the "Smaller" sphere is the same as that of the larger sphere, the electric field reduces to that kq/r^2, the same as the electric field of a point charge at a distance r away.

The smaller cylinder inside the larger cylinder shares the same charge density as the larger cylinder. This allows us to create a relationship between the charge inside the cylinders, and the radius of the cylinders. If the radius of the smaller cylinder is only half that of the larger one, than the charge enclosed by that cylinder is only one quarter that of the larger cylinder.

Placing several metal objects inside the microwave, we see that sparks are created at the surfaces of these objects. These objects include steel wook, a fork, and a CD. 

The charge inside a conducting material is zero. All of the excess charge is location on the surface of the sphere.

Unlike a conducting sphere, whose charge lies entirely on the surface, an insulating sphere has charge that is allowed to move throughout the material. Both spheres have the same charge density. Solving for charge density therefore allows you to come up with an expression for an amount of charge inside a sphere of given radius. 

Enclosing a long thing conducting wire in a cylindrical gaussian surface will create a flux directed radialy outward, therefore the flux is due to curved part of the surface rather than the flat ends. Therefore, flux is equal to the electric field caused by the wire multiplied the area of the curved surface. 

The total electric flux through an enclosed surface equals the total charge enclosed in the surface divided by permittivity of free space ((8.854*10^(-12) * C^2)/(N * m^2)). This is equal the component of the electric field produced by the enclosed perpendicular to the surface multiplied by they area of the surface.