Potential Due to a Uniform Sphere Let S be a hollow sphere of radius R with its center

Chapter 16, Problem 48

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Potential Due to a Uniform Sphere Let S be a hollow sphere of radius R with its center at the origin with a uniform mass distribution of total mass m [since S has surface area 4R2, the mass density is = m/(4R2)]. The gravitational potential V (P) due to S at a point P = (a, b, c) is equal to G S dS (x a)2 + (y b)2 + (z c)2 (a) Use symmetry to conclude that the potential depends only on the distance r from P to the center of the sphere. Therefore, it suffices to compute V (P) for a point P = (0, 0,r) on the z-axis (with r = R). (b) Use spherical coordinates to show that V (0, 0,r) is equal to Gm 4 0 2 0 sin d d R2 + r2 2Rr cos (c) Use the substitution u = R2 + r2 2Rr cos to show that V (0, 0,r) = mG 2Rr |R + r||R r| (d) Verify Eq. (12) for V .