Final answer:
The surface charge density on the inner surface of a metallic spherical shell with charge Q at the center is σi = -Q / (4πR1²), while on the outer surface it is σo = +Q / (4πR2²), where R1 and R2 are the inner and outer radii respectively.
Step-by-step explanation:
When a charge Q is placed at the center of a metallic spherical shell, it induces charges on the inner and outer surfaces of the shell due to electrostatic induction. According to Gauss's law, the electric field inside a conductor in electrostatic equilibrium is zero. Consequently, the charge Q at the center will induce a charge of -Q on the inner surface of the spherical shell to ensure that the electric field within the metal and the cavity is zero.
Therefore, the surface charge density on the inner surface, σi, can be calculated using the formula σi = -Q / (4πR1²). The outer surface, on the other hand, will have a charge of +Q in order to conserve the total charge — the sum of the induced charge on the inner surface and the charge on the outer surface must equal the charge originally placed at the center (Q). Hence, the surface charge density on the outer surface, σo, will be σo = +Q / (4πR2²).
(i) The surface charge density on the inner surface of the shell will be σi = -Q / (4πR1²).
(ii) The surface charge density on the outer surface of the shell will be σo = +Q / (4πR2²).