1. **Problem statement:** A spherical conducting shell has a net charge of +20 C. A point charge of -10 C is placed at the center of the shell. We need to find the charge on the (a) inner surface and (b) outer surface of the shell using Gauss' law. 2. **Relevant principle:** Gauss' law states that the net electric flux through a closed surface is proportional to the net charge enclosed by that surface: $$\Phi_E = \frac{Q_{enc}}{\epsilon_0}$$ For a conductor in electrostatic equilibrium, the electric field inside the conducting material is zero, so charges reside on the surfaces. 3. **Step (a) - Charge on the inner surface:** The point charge at the center is -10 C. To ensure the electric field inside the conductor is zero, the inner surface must have a charge that exactly cancels this charge. Therefore, the charge on the inner surface is: $$Q_{inner} = +10$$ 4. **Step (b) - Charge on the outer surface:** The total charge on the shell is +20 C. Since the inner surface has +10 C, the outer surface must have the remaining charge: $$Q_{outer} = Q_{total} - Q_{inner} = 20 - 10 = 10$$ 5. **Summary:** - Charge on inner surface: $+10$ C - Charge on outer surface: $+10$ C This distribution ensures the electric field inside the conductor is zero and the total charge is conserved.