A thin spherical shell with radius R1 R 1 = 3.00 cm c m is concentric with a larger thin spherical shell with radius 7.00 cm c m . Both shells are made of insulating material. The smaller shell has charge q1=+6.00nC q 1 = + 6.00 n C distributed uniformly over its surface, and the larger shell has charge q2=?9.00nC q 2 = ? 9.00 n C distributed uniformly over its surface. Take the electric potential to be zero at an infinite distance from both shells.
Answer
Electric Potential Due to Concentric Charged Spherical Shells
We are given two spherical shells made of insulating material:
Radius of outer shell, R₂ = 7.00 cm = 0.0700 m
Charge on inner shell, q₁ = +6.00 nC = 6.00 × 10⁻⁹ C
Charge on outer shell, q₂ = −9.00 nC = −9.00 × 10⁻⁹ C
Reference potential: V = 0 at infinity
📘 Step 1: Electric Potential Due to a Spherical Shell
The potential due to a spherical shell at a distance r from its center is:
- Outside the shell (r ≥ R): \( V = \dfrac{1}{4\pi\varepsilon_0} \cdot \dfrac{q}{r} \)
- On or inside the shell (r ≤ R): \( V = \dfrac{1}{4\pi\varepsilon_0} \cdot \dfrac{q}{R} \) (constant)
🔍 Step 2: Total Potential at Any Point
Since the shells are concentric, the total potential at any point is the algebraic sum of potentials due to both shells:
🔢 Step 3: Potential in Different Regions
1️⃣ Region I: r < R₁ (Inside Inner Shell)
Potential is constant due to both shells:
= 9 × 10⁹ × [200 − 128.57] × 10⁻⁹
= 9 × 10⁹ × 71.43 × 10⁻⁹ = 642.87 V
2️⃣ Region II: R₁ < r < R₂ (Between Shells)
Potential due to inner shell acts like a point charge, outer shell is still constant:
3️⃣ Region III: r > R₂ (Outside Both Shells)
Total charge acts like a point charge at center:
✅ Final Answer for Region I:
Note: Since the material is insulating, the charge remains fixed on each shell and does not redistribute based on induction.
Full Explanation for How to Calculate Electric Field of a Spherical Shell