Integrate the vector field A = ze z over a sphere with radius a, centered at the
Question:
Integrate the vector field A = zez over a sphere with radius a, centered at the origin of the Cartesian coordinate system (i.e., compute A · d∑).
(a) Introduce spherical polar coordinates on the sphere, and construct the vectorial integration element d∑ from the two legs adθ eθ̂ and a sin θd∅ e∅̂. Here eθ̂ and e∅̂ are unit-length vectors along the θ and ∅ directions. (Here as in Sec. 1.6 and throughout this book, we use accents on indices to indicate which basis the index is associated with: hats here for the spherical orthonormal basis, bars in Sec. 1.6 for the barred Cartesian basis.) Explain the factors adθ and a sinθd∅ in the definitions of the legs. Show that
(b) Using z = a cos θ and ez = cos θer̂ − sinθeθ̂ on the sphere (where er̂ is the unit vector pointing in the radial direction), show that
(c) Explain why (er̂ , eθ̂ , e∅̂) = 1.
(d) Perform the integral ∫A · d∑ over the sphere’s surface to obtain your final answer (4π/3)a3. This, of course, is the volume of the sphere. Explain pictorially why this had to be the answer.
Step by Step Answer:
Modern Classical Physics Optics Fluids Plasmas Elasticity Relativity And Statistical Physics
ISBN: 9780691159027
1st Edition
Authors: Kip S. Thorne, Roger D. Blandford