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A metallic sphere of radius R = 18 cm is illuminated with light of wavelength lambda = 4000 angstrom. The threshold wavelength for the sphere's material is lambda₀ = 6000 angstrom. A capacitor of capacitance C₀ = 2 microfarad is connected to the sphere by closing a switch. After a long time, what is the charge (in microfarads * volt = microcoulombs) stored on the capacitor? Use hc/e = 12 * 10⁻⁷ V * m.
- 2 microcoulombs
- 1 microcoulomb
- 3 microcoulombs
- 4 microcoulombs
Correct answer: 2 microcoulombs
Solution
The stopping potential is Vₛ = (hc/e)(1/lambda - 1/lambda₀) = 12*10⁻⁷ * (1/(4*10⁻⁷) - 1/(6*10⁻⁷)) = 12*10⁻⁷ * (2.5*10⁶ - 1.667*10⁶) = 12*10⁻⁷ * (8.33*10⁵) = 1 V. The sphere's own capacitance is C_sphere = 4*pi*epsilon₀*R = R/k = 0.18/(9*10⁹) = 2*10⁻¹¹ F = 0.00002 microfarad, which is negligible compared to C₀ = 2 microfarad. After a long time, the capacitor and sphere reach equilibrium potential Vₛ = 1 V. Charge on capacitor = C₀ * Vₛ = 2 * 10⁻⁶ * 1 = 2 * 10⁻⁶ C = 2 microcoulombs.
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