A 5.0-nC point charge is embedded at the center of a nonconducting sphere (radius = 2.0 cm) which has a charge of -8.0 nC distributed uniformly throughout its volume. What is the magnitude of the electric field at a point that is 1.0 cm from the center of the sphere

3.6 × 10⁵ N/C = 360 kN/C

Explanation:

Let R = 2.0 cm be the radius of the sphere and q = -8.0 nC be the charge in it. Let q₁ be the charge at radius r = 1.0 cm. Since the charge is uniformly distributed, the volume charge density is constant. So, q/4πR³ = q₁/4πr³

q₁ = q(r/R)³. The electric field due to q₁ at r is E₁ = kq₁/r² = kq(r/R)³/r² = kqr/R³

The electric field due to the point charge q₂ = 5.0 nC is E₂ = kq₂/r².

So, the magnitude of the total electric field at r = 1.0 cm is

E = E₁ + E₂ = kqr/R³ + kq₂/r² = k(qr/R³ + q₂/r²)

E = 9 × 10⁹(-8 × 10⁻⁹ C × 1 × 10⁻² m/(2 × 10⁻² m)³ + 5 × 10⁻⁹ C/(1 × 10⁻² m)²)

E = 9 × 10⁹(-1 × 10⁻⁵ + 5 × 10⁻⁵)

E = 9 × 10⁹(4 × 10⁻⁵)

E = 36 × 10⁴ N/C = 3.6 × 10⁵ N/C = 360 kN/C