An isolated charged conducting sphere of radius 12.0 cm creates an electric field of 4.90 ×10^4

Step 1: Determine the charge on the sphere using the electric field formula. The electric field

An isolated charged conducting sphere of radius 12.0 cm...

An isolated charged conducting sphere of radius $12.0 \mathrm{cm}$ creates an electric field of $4.90 \times 10^{4} \mathrm{N} / \mathrm{C}$ at a distance $21.0 \mathrm{cm}$ from its center. (a) What is its surface charge density? (b) What is its capacitance?

Key Concepts

Gauss's Law

Gauss's Law relates the net electric flux through a closed surface to the charge enclosed by that surface. In the context of a charged conducting sphere, this law simplifies the calculation of the electric field by allowing the sphere to be treated as a point charge when calculating the field outside the sphere. This principle is fundamental in determining the relationship between the measured electric field and the total charge present on the sphere.

Electric Field of a Conducting Sphere

For a spherical conductor, the electric field outside the sphere behaves as if the entire charge were concentrated at the center. This result, derived from Gauss's Law, implies that the electric field at any point outside the sphere can be calculated using Coulomb's law, making it possible to infer the total charge from a known field at a given distance.

Surface Charge Density

Surface charge density is defined as the amount of electric charge per unit area on the surface of a conductor. For a sphere, it is calculated by dividing the total charge on the sphere by the surface area (which is 4?R² for a sphere of radius R). This concept is crucial for understanding how the charge distributes itself over the conductor's surface in electrostatic equilibrium.

Capacitance of a Sphere

The capacitance of an isolated conducting sphere is a measure of its ability to store charge per unit potential. It is given by the formula C = 4???R, where ?? is the permittivity of free space and R is the radius of the sphere. This definition is important in electrostatics as it provides a direct relationship between the geometry of a conductor and its electrical properties.

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