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A solid metallic sphere of radius a carries a charge of +3Q. A concentric conducting spherical shell of radius b (b > a) carries a charge of -Q. What is the value of x if the electric field at a distance R from the centre (a < R < b) equals x*Q / (8 * pi * epsilon₀ * R²)?
- 3
- 6
- 12
- 2
Correct answer: 6
Solution
For a < R < b, the Gaussian surface encloses only the inner solid metallic sphere with charge +3Q (the shell's charge is on its surfaces, but between a and b only the inner sphere's charge matters; the inner surface of the shell has -3Q induced, but R < b so we're inside the shell material — wait, the shell extends from inner radius to outer radius; for a conducting shell, field inside conductor is 0 and Gaussian surface at radius R where a < R < b is in the region between the sphere and the shell). Enclosed charge = +3Q. By Gauss's law: E * 4*pi*R² = 3Q/epsilon₀. E = 3Q/(4*pi*epsilon₀*R²) = 6Q/(8*pi*epsilon₀*R²). So x = 6.
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