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A total charge Q is distributed uniformly throughout the material of a hollow sphere having inner radius r1 and outer radius r2. Determine the electric field at a point located at distance x from the center, where r1 < x < r2.
- Q*(x³ - r1³) / (4*pi*epsilon0*x²*r2³)
- Q*(x³ - r1³) / (4*pi*epsilon0*x²*(r2³ - r1³))
- Q*x / (4*pi*epsilon0*(r2³ - r1³))
- Q*(x³ - r1³) / (4*pi*epsilon0*r1²*(r2² - r1²))
Correct answer: Q*(x³ - r1³) / (4*pi*epsilon0*x²*(r2³ - r1³))
Solution
Since charge is uniformly distributed in the shell material, the volume charge density is rho = Q / [volume of shell material] = Q / [(4/3)*pi*(r2³ - r1³)]. For a Gaussian sphere of radius x, the enclosed charge is rho times the volume from r1 to x. By Gauss's law, E * 4*pi*x² = Q_enc / epsilon0.
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