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. (Figure 1) 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.

Part A

What is the electric flux Φ3Φ3Phi_3 through the annular ring, surface 3?

Express your answer in terms of CCC, r1r1r_1, r2r2r_2, and any constants.

Part B

What is the electric flux Φ1Φ1Phi_1 through surface 1?

Express Φ1Φ1Phi_1 in terms of CCC, r1r1r_1, r2r2r_2, and any needed constants.

Part C

What is the electric flux Φ2Φ2 passing outward through surface 2?

Express Φ2Φ2Phi_2 in terms of r1r1r_1, r2r2r_2, CCC, and any constants or other known quantities.