The electric flux may not be uniform throughout a particular region of space. We can determine the total electric flux by examining a portion of the electric flux passing through a small segment of the total area. If we look at infinitesimal sections of the area we get: A surface integral is evaluated over the specific surface examined We typically integrate over closed surfaces and therefore use this form of the expression.

Let us look at the electric flux through a thin spherical shell surrounding a point charge. (sign on charge doesn’t matter) E and A are parallel at all points on the sphere E doesn’t change with the surface area of the spherical shell (all parts of A are the same distance from the point charge). surface area of a sphere Constant! The electric flux through any closed surface surrounding a charged object is constant!

If a charge is placed outside a closed surface the electric flux is zero since the number of electric field lines entering the surface is equal to the number of electric field lines leaving the surface. We now have a new method for determining the electric field through any closed surface. Gauss’s Law q enclosed is the charge enclosed by the surface We can also use this equation to find the electric field strength at any point by surrounding the charged object with an imaginary closed surface called a Gaussian Surface. The Gaussian surface can be of any shape, but you should choose a shape appropriate for the particular charge distribution you are examining.

Gaussian Surfaces Sphere Cylinder

A cylindrical piece of insulating material is placed in an external electric field, as shown. The net electric flux passing through the surface of the cylinder is 1. positive. 2. negative. 3. zero.

Example: An insulating solid sphere with a uniform volume charge distribution has a total charge Q and a radius a. a)Determine an expression for the electric field inside the sphere. b)Determine an expression for the electric field outside the sphere. a) E and A are parallel E is not a function of A E is linear with radial position!