Concentric Spherical Charge Distributions with Varying Charge Density

To determine which of the statements are correct, we first calculate the total charge enclosed by region $A$ and region $B$.

Step 1: Charge inside region $A$ ($Q_A$)

Region $A$ is a sphere of radius $r_A = 1$ with a volumetric charge density $\rho_A = kr$. Using spherical shells of radius $r$ and thickness $dr$: $Q_A = \int_0^{r_A} \rho_A(r) \cdot 4\pi r^2 \, dr = \int_0^1 (kr) \cdot 4\pi r^2 \, dr$ $Q_A = 4\pi k \int_0^1 r^3 \, dr = 4\pi k \left[ \frac{r^4}{4} \right]_0^1 = \pi k$

Step 2: Charge inside region $B$ ($Q_B$)

Region $B$ is a spherical shell extending from $r_A = 1$ to outer radius $r_B$, with charge density $\rho_B = \frac{2k}{r}$: $Q_B = \int_{r_A}^{r_B} \rho_B(r) \cdot 4\pi r^2 \, dr = \int_1^{r_B} \left(\frac{2k}{r}\right) \cdot 4\pi r^2 \, dr$ $Q_B = 8\pi k \int_1^{r_B} r \, dr = 8\pi k \left[ \frac{r^2}{2} \right]_1^{r_B} = 4\pi k (r_B^2 - 1)$

Step 3: Total Charge of the Configuration ($Q_{\text{total}}$)

$Q_{\text{total}} = Q_A + Q_B = \pi k + 4\pi k(r_B^2 - 1) = \pi k (4r_B^2 - 3)$

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Step 4: Evaluation of Options

• Option (A): For the electric field outside region $B$ to be zero everywhere, the net enclosed charge $Q_{\text{total}}$ must be zero: $Q_{\text{total}} = 0 \implies \pi k (4r_B^2 - 3) = 0 \implies r_B = \frac{\sqrt{3}}{2}$ Since $r_B$ must be greater than $r_A = 1$, $r_B = \frac{\sqrt{3}}{2} \approx 0.866$ is physically impossible. Furthermore, substituting $r_B = \sqrt{\frac{3}{2}}$ into $Q_{\text{total}}$ gives: $Q_{\text{total}} = \pi k \left(4 \left(\frac{3}{2}\right) - 3\right) = 3\pi k \neq 0$ Therefore, the electric field is non-zero outside $B$. Option (A) is incorrect.

• Option (B): If $r_B = \frac{3}{2}$, the total charge is: $Q_{\text{total}} = \pi k \left( 4 \left(\frac{3}{2}\right)^2 - 3 \right) = \pi k (9 - 3) = 6\pi k$ The electric potential $V$ just outside $B$ (at $r = r_B = \frac{3}{2}$) is: $V = \frac{Q_{\text{total}}}{4\pi \epsilon_0 r_B} = \frac{6\pi k}{4\pi \epsilon_0 \left(\frac{3}{2}\right)} = \frac{6\pi k}{6\pi \epsilon_0} = \frac{k}{\epsilon_0}$ Option (B) is correct.

• Option (C): If $r_B = 2$, the total charge of the configuration is: $Q_{\text{total}} = \pi k (4(2)^2 - 3) = \pi k (16 - 3) = 13\pi k$ The option states that the total charge is $15\pi k$. Option (C) is incorrect.

• Option (D): If $r_B = \frac{5}{2}$, the total charge is: $Q_{\text{total}} = \pi k \left( 4 \left(\frac{5}{2}\right)^2 - 3 \right) = 22\pi k$ The magnitude of the electric field $E$ just outside $B$ is: $E = \frac{Q_{\text{total}}}{4\pi \epsilon_0 r_B^2} = \frac{22\pi k}{4\pi \epsilon_0 \left(\frac{5}{2}\right)^2} = \frac{22 k}{25 \epsilon_0} \neq \frac{13\pi k}{\epsilon_0}$ Option (D) is incorrect.

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Conclusion

The correct statement is (B).