The problem describes a charged metallic sphere surrounded by a dielectric spherical shell. Here's how we approach each part:

Given Information

1. Charge on metallic sphere: $Q = -5 \, \text{nC}$
2. Radius of the metallic sphere: $a = 2 \, \text{cm}$
3. Dielectric constant: $\varepsilon_r = 35$
4. Inner radius of the dielectric shell: $a = 2 \, \text{cm}$
5. Outer radius of the dielectric shell: $b = 6 \, \text{cm}$

(a) Potential at the center of the metallic sphere

The potential $V$ at the surface of a metallic sphere is uniform throughout the conductor. Using the formula for the potential due to a spherical shell system:

$V = \frac{1}{4 \pi \varepsilon_0} \cdot \frac{Q}{a}$

where $\varepsilon_0 = 8.85 \times 10^{-12} \, \text{F/m}$ is the vacuum permittivity.

(b) Polarization surface charge densities

The polarization surface charge densities are given by:

1. Inner surface of dielectric: $\sigma_{\text{inner}} = -P = -\varepsilon_0 \chi_e E$

2. Outer surface of dielectric: $\sigma_{\text{outer}} = \varepsilon_0 \chi_e E$

Here $E$ is the electric field inside the dielectric, related to the charge $Q$.

Would you like me to calculate the exact values or expand on the derivations? Let me know! Below are some guiding questions.

Relative Questions

1. What is the potential at the metallic sphere's outer surface and inside the dielectric material?
2. How does the dielectric constant affect the field and polarization charges?
3. Can you explain how the inner and outer radii of the dielectric affect the electric field?
4. What happens if $Q$ changes its sign or magnitude? How will $\sigma_{\text{inner}}$ and $\sigma_{\text{outer}}$ change?
5. How would the system behave if the dielectric material was not linear (non-linear dielectric)?

Tip

Always remember that in a conductor, the electric field inside is zero, and the potential remains constant.