Since the total charge on the rod (Q) is simply the product of the charge density and the total
length of the rod, this reduces to
This is exactly the expression for the electric field from a point charge. Thus, as you move
farther and farther from the rod, the rod does indeed begin to look like a point charge.
The long, hollow plastic cylinder at right has inner
radius a, outer radius b, and uniform charge density
. Find the electric field at all points in a plane
perpendicular to the cylinder near its midpoint.
For certain situations, typically ones with a high degree of symmetry, Gauss’ Law allows you
to calculate the electric field relatively easily. Gauss’ Law, mathematically, states:
Let’s describe what this means in English. The left side of the equation involves the vector dot
product between the electric field and an infinitesimally small area that is a piece of a larger
closed surface (termed the gaussian surface). This dot product between electric field and area
is termed electric flux, and is often visualized as the amount of field that “passes through” the
little piece of area. The integral simply tells us to add up all of these infinitesimal electric
fluxes to get the total flux through the entire closed surface.
The gist of Gauss’ Law is that this total electric flux is exactly equal to the total amount of
electric charge enclosed within the gaussian surface, divided by a constant,
permittivity of free space, a constant equal to 8.85 x 10
Somewhat counter intuitively, the key to applying Gauss’ Law is to choose a gaussian surface
such that you never really have to do the integral on the left side of the equation! To try to
help you understand what I’m talking about, let’s walk through the solution of the above
problem. The following sequence of steps will help you understand the process of applying
1. Choose the appropriate gaussian surface.
2. Carefully draw the hypothetical gaussian surface at the location of interest.
3. Carefully draw the electric field at all points on the gaussian surface.
4. Write an expression for the surface area parallel to the electric field.
5. Write an expression for q
, the charge enclosed within the gaussian surface.
Apply Gauss’ Law and determine the electric field at all points on this hypothetical surface.