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 are (a) the volume charge density for the sphere?

Verified step by step guidance

1. Calculate the volume of the sphere using the formula for the volume of a sphere, \( V = \frac{4}{3} \pi R^3 \), where \( R \) is the radius of the sphere.
2. Use Gauss's Law, \( \oint \mathbf{E} \cdot d\mathbf{A} = \frac{Q_{\text{enc}}}{\epsilon_0} \), where \( Q_{\text{enc}} \) is the charge enclosed by the Gaussian surface and \( \epsilon_0 \) is the permittivity of free space. Choose a Gaussian surface that is a sphere with radius \( r \) (8.00 cm in this case) which encloses the entire charge of the sphere.
3. Since the electric field \( E \) is given at the distance \( r \), and the area of the Gaussian surface is \( A = 4\pi r^2 \), the left side of Gauss's Law becomes \( E \cdot 4\pi r^2 \).
4. Solve Gauss's Law for \( Q_{\text{enc}} \) using the expression from step 3 and the known value of \( \epsilon_0 \).
5. Calculate the volume charge density, \( \rho \), using the formula \( \rho = \frac{Q_{\text{enc}}}{V} \), where \( V \) is the volume of the sphere calculated in step 1.

Key Concepts

Electric Field

The electric field is a vector field that represents the force exerted by an electric charge on other charges in its vicinity. It is defined as the force per unit charge and is measured in newtons per coulomb (N/C). For a uniformly charged sphere, the electric field outside the sphere behaves as if all the charge were concentrated at the center, allowing for straightforward calculations using Gauss's law.

Volume Charge Density

Volume charge density, denoted by ρ, is a measure of the amount of electric charge per unit volume within a given region of space. It is expressed in coulombs per cubic meter (C/m³). For a uniformly charged insulating sphere, the volume charge density can be calculated by dividing the total charge by the volume of the sphere, which is given by the formula V = (4/3)πR³.

Gauss's Law

Gauss's Law relates the electric flux through a closed surface to the charge enclosed by that surface. Mathematically, it states that the total electric flux is equal to the enclosed charge divided by the permittivity of free space. This law is particularly useful for calculating electric fields in symmetric charge distributions, such as spheres, as it simplifies the process of finding the electric field at a distance from the charge.