1.3 Integral Calculus (iv) 1 ZA 1 (v) (ii) (vi) FIGURE 1.29 1 So much for the left side of the divergence theorem. To evaluate the surface integral we must consider separately the six faces of the cube: (i) (ii) (iii) (iv) (v) (vi) So the total flux is: 1 1 [ v · da = f' f' y²dy dz = } [ v. da v.da [ v · da = L 1 1 - S² S² y² dy dz = − } . [ = Sv. $ S v.da L 1 1 (2x + z²) dx dz = 3. 1 - S' S² ² dx d² = − } . L 1 1 [ v-da = L'S' dz S' S² 2y dx dy = 1. Sv da = - S' S L L · ½ 1 0dx dy = 0. v da = -+-+1+0=2, as expected. 33 33 Example 1.10. Check the divergence theorem using the function v = y²â+ (2xy + z²) ŷ + (2yz) z and a unit cube at the origin (Fig. 1.29). Solution In this case V.v=2(x + y), and 1 {2(x + y) dt = 2 f f f (x + y)dx dy dz. 1 S' L' = (x + y) dx = + y, = 1+y, f² (4 + y) dy = 1, [ '1dz = 1. 0 Thus, V.vdt = 2.

QEX. Need help with part B. Need theory and full detailed solution.

A 93.0 A current circulates around a 2.30-mm-diameter superconducting ring. Y Part A What is the ring's magnetic dipole moment? Express your answer in amperes times meter squared. ▸ View Available Hint(s) JAM Submit Part B 195] ΑΣΦΑ R What is the on-axis magnetic field strength 5.90 cm from the ring? Express your answer with the appropriate units. View Available Hint(s) Valua ? Unite A-m

For multipole expansion, under what conditions, if any, is it an exact expression, and under what conditions, if any, is it an approximation? In what limits/cases is it the most useful?