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Next Problem List Previous (1 point) B = (0,0, b) 1 A uniform magnetic field B has constant strength b teslas in the z-direction [ie. (a, y, 0 Bxr is a vector potential for B, where r (a) Verity that A (b) Use the Stokes theorem to calculate the flux of B through the rectangle with vertices A, B. C, and D in Figure 17 C F FIGURE 17 A (8,0,7), B (8,5,0), C (0,5,0), D (0,0,7), F (8,0,0) Flux(B) Preview My Answers Submit Answers

Answer 1

We need to verify that nabla times A = B, we have Nabla times A = 1/2 nabla times (B times r) and, B times r = det (i 0 x j 0 y k b 0) = -byi + bxj Therefore nabla times A = 1/2 nabla times (B times r) = 1/2 nabla times (-by br 0) = 1/2 det (i Partial/Partial x - by j Partial/Partial y bx k Partial/Partial z 0) = 1/2 (0, 0, b + b) = (0, 0, b) = B Therefore A = 1/2 (B times r) is vector potential of B. b) We need to find flux (B) = Integral_R B. ds = Integral_R (nabla times A) ds Where R is given rectangle & ds the surface element. Now by stokes theorem, we have Flux (B) = Integral_R (nabla times A) middot ds = Integral_R A middot ds Here ds is