You want to find the dot product between two vectors, \(\vec{A}\) and \(\vec{B}\text{.}\) What do you know about the directions of the vectors if the dot product is positive? What if the dot product is negative? What if the dot product is zero?
The double-integral sign indicates that the integral is over a two-dimensional surface, and the limits of integration must be chosen so that they cover the entirety of the desired surface. The differential area vector \(d\vec{A}\) represents an infinitesimal piece of the total surface.
Flux can be described as how much of something flows perpendicularly through an area. Common examples of flux might be how much water flows through the surface of your faucet, how many photons travel through the opening of a telescope, or how much wind flows through a window.
The electric flux tells us about the total electric field passing perpendicularly through a surface. While the electric field is not actually “flowing” through a surface, it is still useful to discuss how much of the electric field is passing through a surface. The integral above is called a surface integral, and can be thought of as summing the electric flux passing through each infinitesimal \(d\vec{A}\) that makes up a macroscopic surface area.
Suppose the book were placed in a region with a uniform electric field that points upward: \(\vec{E} = E_o\hat{y}\text{.}\) If the area of the front cover is \(A\text{,}\) determine the electric flux through the front cover. What happens to the integral in a case like this?
Suppose you know the electric field everywhere in space and your surface of interest is a sphere centered on the origin. What is the area vector for this surface? Outline a general procedure for finding the electric flux through this surface. Can you think of any electric fields for which this procedure will be very challenging? Very easy?
