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

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Question:

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

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Answer:3.6 × 10⁵ N/C = 360 kN/CExplanation: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 = q₁/4πr³q₁ = q(r/R)³. The electric field due to q₁ at r is E₁ = kq₁/r² = kq(r/R)³/r² = kqr/R³The 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 isE = 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

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