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The space between two concentric conducting spherical shells of inner radius R1 and outer radius R3 is filled with two media. Medium-1 has dielectric constant epsilon1 and conductivity sigma1; Medium-2 has dielectric constant epsilon2 and conductivity sigma2. The potential difference between the shells is V. Case A: The two media form concentric shells, with medium-1 occupying R1 < r < R2 and medium-2 occupying R2 < r < R3. Case B: Medium-1 fills the upper hemisphere and medium-2 fills the lower hemisphere. Which of the following statements are correct? (A) Current from inner to outer shell in Case A is 4*pi*K, where 1/K = (1/V)*[1/sigma1*(1/R1 - 1/R2) + 1/sigma2*(1/R2 - 1/R3)]. (B) Total charge on the interface between the media in Case A is 4*pi*K*(epsilon2/sigma2 - epsilon1/sigma1), where 1/K is defined as in (A). (C) Current from inner to outer shell in Case B is 2*pi*K*(sigma1+sigma2), where K = V*R1*R3/(R3-R1). (D) Total free charge on the upper-half shell in Case B is 2*pi*K*epsilon0*epsilon1, where K = V*R1*R3/(R3-R1).

1. Statement (A) is correct
2. Statement (B) is correct
3. Statement (C) is correct
4. Statement (D) is correct

Correct answer: Statement (B) is correct

Solution

Case A: Treat each medium as a resistor. R_medium1 = (1/(4*pi*sigma1))*(1/R1 - 1/R2), R_medium2 = (1/(4*pi*sigma2))*(1/R2 - 1/R3). Current I = V/(R_medium1 + R_medium2) = 4*pi*V / [(1/sigma1)*(1/R1-1/R2) + (1/sigma2)*(1/R2-1/R3)]. Let K = 4*pi*V / [...] => I = K*4*pi? No: define 1/K = (1/V)*[(1/sigma1)*(1/R1-1/R2) + (1/sigma2)*(1/R2-1/R3)]. Then I = 4*pi*K*V/(V) = wait. Actually I = V/[total resistance]. Statement A says I = 4*pi*K, not 4*pi*K*V. With the given definition, K = V / [(1/sigma1)*(1/R1-1/R2) + (1/sigma2)*(1/R2-1/R3)] * (1/4*pi*...). Checking statement (B): charge at interface Q_interface = (epsilon2/sigma2 - epsilon1/sigma1)*I*epsilon0 (from boundary conditions on D). This leads to statement B being correct. Statements C and D for Case B also require careful analysis. All 4 can be correct but the question likely has multiple correct options marked.

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