A spherically symmetric charge distribution produces the electric field E (→ above E) = (5000r²) rˆ N/C, where r is in m. (c) How much charge is inside this 40-cm-diameter spherical surface?

Verified step by step guidance

1. Identify the given electric field equation, E(→) = (5000r²) rˆ N/C, and recognize that rˆ is the radial unit vector.
2. Recall Gauss's Law, which states that the electric flux through a closed surface is equal to the charge enclosed by the surface divided by the permittivity of free space, ε₀. Mathematically, it is expressed as \( \Phi_E = \oint \vec{E} \cdot d\vec{A} = \frac{Q_{\text{enc}}}{\epsilon_0} \).
3. Calculate the electric flux, \( \Phi_E \), through the spherical surface. Since the electric field is radial and the area vector is also radial, the dot product simplifies to \( E \cdot A \).
4. Compute the surface area, A, of the sphere using the formula \( A = 4\pi r^2 \), where r is the radius of the sphere. Note that the diameter is given, so the radius is half of the diameter.
5. Substitute the values of \( \Phi_E \) and \( \epsilon_0 \) into Gauss's Law to solve for the enclosed charge, \( Q_{\text{enc}} \).

Key Concepts

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 principle is crucial for analyzing electric fields produced by symmetric charge distributions, allowing us to simplify calculations by considering only the charge within a defined boundary.

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 can vary with distance from the charge. In this problem, the electric field is given as a function of the radial distance, indicating how the field strength changes with distance from the center of the charge distribution.

Spherical Symmetry

Spherical symmetry refers to a situation where a physical quantity, such as charge distribution or electric field, is uniform in all directions from a central point. This symmetry simplifies the analysis of electric fields and potentials, as it allows the use of spherical coordinates and the application of Gauss's Law. In this case, the charge distribution's spherical symmetry means that the electric field can be treated as a function of the radial distance alone.