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A spherical capacitor has inner radius a and outer radius b. One half of the space between the spheres is filled with a dielectric of dielectric constant k = 2 and electrical conductivity sigma, while the other half is vacuum. The capacitor is given an initial charge Q0. Due to the conductivity of the dielectric, charge leaks through it. Which of the following statements is/are correct?

1. The time constant for the discharge is 3*epsilon0 / sigma
2. The time constant for the discharge is 5*epsilon0 / sigma
3. The resistance of the dielectric half-shell is (b-a) / (2*pi*sigma*a*b)
4. The resistance of the dielectric half-shell is (b-a) / (4*pi*sigma*a*b)

Correct answer: The time constant for the discharge is 3*epsilon0 / sigma

Solution

The capacitor has two halves in parallel: a vacuum half with capacitance C1 = 2*pi*epsilon0*a*b/(b-a) and a dielectric half with capacitance C2 = 2*k*pi*epsilon0*a*b/(b-a) = 4*pi*epsilon0*a*b/(b-a). The resistance of the dielectric half is R = (b-a)/(2*pi*sigma*a*b). The time constant tau = R*(C1+C2) = 3*epsilon0/sigma.

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