± Calculating Flux for Hemispheres of Different Radii LearningGoal: To understand the definition of electric flux, and how tocalculate it. Flux is the amount of a vector field that \"flows\"through a surface. We now discuss the electric flux through asurface (a quantity needed in Gauss's law): ΦE=∫E⃗ ⋅dA⃗ , where ΦEis the flux through a surface with differential area element dA⃗ ,and E⃗ is the electric field in which the surface lies. There areseveral important points to consider in this expression: It is anintegral over a surface, involving the electric field at thesurface. dA⃗ is a vector with magnitude equal to the area of aninfinitesmal surface element and pointing in a direction normal(and usually outward) to the infinitesmal surface element. Thescalar (dot) product E⃗ ⋅dA⃗ implies that only the component of E⃗normal to the surface contributes to the integral. That is, E⃗ ⋅dA⃗=|E⃗ ||dA⃗ |cos(θ), where θ is the angle between E⃗ and dA⃗ . Whenyou compute flux, try to pick a surface that is either parallel orperpendicular to E⃗ , so that the dot product is easy to compute.(Figure 1) Two hemispherical surfaces, 1 and 2, of respective radiir1 and r2, are centered at a point charge and are facing each otherso 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 |, andr^=r⃗ /r is the unit vector in the radial direction.