Let's break down the problem step by step: ### **Given Data** - Solid metal sphere (radius = 8 cm) - Charge, \( Q_1 = 10.13 \) mC \( = 10,130 \) μC - First metallic shell (radius = 15 cm) - Charge, \( Q_2 = -1.52 \) mC \( = -1,520 \) μC - Larger metallic shell (inner radius = 20 cm, outer = 24 cm) - All shells and the sphere are **concentric**. --- ## **Step 1: Charge on the Inner Wall of the Larger Shell** Let's call: - **Sphere charge:** \( Q_1 = 10,130 \) μC - **First shell total charge:** \( Q_2 = -1,520 \) μC - **Larger shell:** Unknown charges on inner (\( Q_{a, \text{in}} \)) and outer (\( Q_{a, \text{out}} \)) walls ### **a. What is the charge on the inner wall of the larger shell?** #### **Principle:** The electric field inside a conductor in electrostatic equilibrium is zero. **All charges inside the cavity up to the inner wall (20 cm) must be neutralized by the inner wall.** #### **Total charge enclosed within 20 cm:** - Sphere: \( Q_1 = 10,130 \) μC - First shell: \( Q_2 = -1,520 \) μC \[ Q_{\text{enclosed}} = Q_1 + Q_2 = 10,130 \, \mu\text{C} + (-1,520) \, \mu\text{C} = 8,610 \, \mu\text{C} \] #### **The inner wall must have a charge of:** \[ Q_{a, \text{in}} = -Q_{\text{enclosed}} = -8,610 \, \mu\text{C} \] --- ## **Step 2: Charge on the Outer Wall if the Field Outside is Zero** ### **b. If the electric field outside the larger shell is zero, what is the charge on the outer wall?** If the field outside is zero, the **total charge** enclosed by the outer shell must be zero. #### **Total charge inside (including both walls of shells and sphere):** - Sphere: \( Q_1 = 10,130 \) μC - First shell: \( Q_2 = -1,520 \) μC - Inner wall of large shell: \( Q_{a, \text{in}} = -8,610 \) μC - Outer wall of large shell: \( Q_{a, \text{out}} = ? \) Set total charge to zero: \[ Q_1 + Q_2 + Q_{a, \text{in}} + Q_{a, \text{out}} = \] \[ 10,130 + (-1,520) + (-8,610) + Q_{a, \text{out}} = \] \[ 10,130 - 1,520 - 8,610 + Q_{a, \text{out}} = \] \[ Q_{a, \text{out}} = 1,520 \, \mu\text{C} \] --- ## **Final Answers** ### **a. Charge on the inner wall of the larger shell:** \[ \boxed{-8,610 \, \mu\text{C}} \] ### **b. Charge on the outer wall of the larger shell:** \[ \boxed{1,520 \, \mu\text{C}} \]