Charge $q$ is distributed uniformly throughout the volume of an insulating sphere of radius $R=4.00$ cm. At a distance of $r=8.00$ cm from the center of the sphere, the electric field due to the charge distribution has magnitude $E=940$ N/C. What is the volume charge density for the sphere?

Verified step by step guidance

1. Understand that the problem involves a uniformly charged insulating sphere, and we need to find the volume charge density \( \rho \). The electric field \( E \) is given at a point outside the sphere.
2. Recall Gauss's Law, which states that the electric field \( E \) due to a symmetric charge distribution can be calculated using \( E = \frac{Q}{4\pi\varepsilon_0 r^2} \), where \( Q \) is the total charge, \( \varepsilon_0 \) is the permittivity of free space, and \( r \) is the distance from the center of the sphere.
3. Since the electric field is given at \( r = 8.00 \) cm, which is outside the sphere, the entire charge \( Q \) of the sphere contributes to the electric field at this point.
4. The volume charge density \( \rho \) is defined as \( \rho = \frac{Q}{V} \), where \( V \) is the volume of the sphere. The volume \( V \) of the sphere can be calculated using the formula \( V = \frac{4}{3}\pi R^3 \), where \( R \) is the radius of the sphere.
5. To find \( \rho \), first solve for \( Q \) using the electric field equation: \( Q = E \cdot 4\pi\varepsilon_0 r^2 \). Then, substitute \( Q \) into the volume charge density formula \( \rho = \frac{Q}{V} \) to find \( \rho \).

Key Concepts

Electric Field

The electric field is a vector field around a charged object where a force would be exerted on other charges. It is defined as the force per unit charge and is measured in newtons per coulomb (N/C). In this problem, the electric field at a distance from the sphere is given, which helps in determining the charge distribution.

Gauss's Law

Gauss's Law relates the electric flux through a closed surface to the charge enclosed by that surface. It is expressed as Φ = Q_enclosed/ε₀, where Φ is the electric flux, Q_enclosed is the charge within the surface, and ε₀ is the permittivity of free space. This law is crucial for calculating the electric field due to symmetric charge distributions like spheres.

Volume Charge Density

Volume charge density (ρ) is the measure of electric charge per unit volume within a region. It is expressed in coulombs per cubic meter (C/m³). For a uniformly charged sphere, the volume charge density can be calculated by dividing the total charge by the sphere's volume, which is essential for understanding how charge is distributed within the sphere.