± Calculating Flux for Hemispheres of Different Radii Learning Goal:
To understand the definition of electric flux, and how to calculate it. Flux is the amount of a vector field that "flows" through a surface. We now discuss the electric flux through a surface (a quantity needed in Gauss's law): ΦE=∫E⃗ ⋅dA⃗ , where ΦE is the flux through a surface with differential area element dA⃗ , and E⃗ is the electric field in which the surface lies. There are several important points to consider in this expression: It is an integral over a surface, involving the electric field at the surface. dA⃗ is a vector with magnitude equal to the area of an infinitesmal surface element and pointing in a direction normal (and usually outward) to the infinitesmal surface element. The scalar (dot) product E⃗ ⋅dA⃗ implies that only the component of E⃗ normal to the surface contributes to the integral. That is, E⃗ ⋅dA⃗ =|E⃗ ||dA⃗ |cos(θ), where θ is the angle between E⃗ and dA⃗ . When you compute flux, try to pick a surface that is either parallel or perpendicular to E⃗ , so that the dot product is easy to compute.
Two hemispherical surfaces, 1 and 2, of respective radii r1 and r2, are centered at a point charge and are facing each other so that their edges define an annular ring (surface 3), as shown. The field at position r⃗ due to the point charge is: E⃗ (r⃗ )=Cr2r^ where C is a constant proportional to the charge, r=|r⃗ |, and r^=r⃗ /r is the unit vector in the radial direction.