A spherical, non-conducting shell of inner radius r1= 10 cm and outer radius r2= 15 cm carries a total charge Q = 15 μC distributed uniformly throughout its volume. What is the electric field at a distance r= 12 cm from the center of the shell?

E = 2.87 × 10⁶ N/C

Explanation:

Electric Field Intensity Formula  E = K q / r²

We will first find the charge density according the given data. (After that we will find charge enclosed by r=12 cm)

r₁ = 10 cm = 0.1 m, r₂ = 15 cm = 0.15 m, Q = 15 μC =15 × 10⁻⁶C

charge density = Charge enclosed / volume

Volume shell = 4/3 π (r₂³ - r₁³) = 4/3 π ( 0.15³ - 0.10³) =9.948 × 10⁻³ m³

charge density = 15 × 10⁻⁶C / 9.948 × 10⁻³ m³ = 1.51 × 10⁻³ C /m³

Now the Volume of shell at (r=12 cm = .12 m)  = 4/3 π ( 0.12³ - 0.10³) = 3.049 × 10⁻³ m³

Now the amount of charge inside r=0.12 m which is

Charge = Charge density × volume = 0.151 × 10⁻³ C /m³ ×  3.049 × 10⁻³ m³

=4.60 × 10⁻⁶ C

So E = 8.99 × 10 ⁹ N m²/C × 4.60 × 10⁻⁶ C / (0.12 m)² = 2.87 × 10⁶ N/C

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