A large metal sphere has four times the diameter of a...

Name: Problem 30, Chapter 22: Essential University Physics Global Edition
Uploaded: 2023-03-18T13:47:13-08:00
Duration: 3 min 11 s
Channel: Vishal Gupta
Description: A large metal sphere has four times the diameter of a smaller sphere and carries four times the charge. Both spheres are isolated, so their surface charge densities are uniform. Compare (a) the potentials (relative to infinity) and (b) the electric field strengths at their surfaces.

A large metal sphere has four times the diameter of a smaller sphere and carries four times the charge. Both spheres are isolated, so their surface charge densities are uniform. Compare (a) the potentials (relative to infinity) and (b) the electric field strengths at their surfaces.

Key Concepts

Uniform Surface Charge Density

For conductive objects that are isolated from other conductive bodies, the charge tends to distribute uniformly over the surface. This uniform distribution is crucial because it allows the use of symmetry in electrostatic calculations, simplifying the analysis of both potential and electric field around the object.

Scaling Laws in Electrostatics

Scaling laws illustrate how altering the physical dimensions of an object and its charge impacts electric quantities such as potential and field strength. For instance, increasing the radius while modifying the charge results in different scaling behaviors for the potential (which scales with 1/R) and the electric field (which scales with 1/R²). This idea is pivotal for comprehending how geometric changes influence electrostatic properties.

Electric Potential of a Charged Sphere

The electric potential at the surface of a charged sphere is given by V = kQ/R, where k is Coulomb's constant, Q is the charge, and R is the radius. This formula demonstrates that the potential depends directly on the amount of charge and inversely on the size of the sphere, making it a key concept in understanding how potential is affected by changes in these parameters.

Electric Field at the Surface of a Charged Sphere

The electric field just outside a uniformly charged sphere can be calculated using E = kQ/R². This relationship shows that the electric field is influenced by both the total charge and the radius of the sphere, falling off with the square of the distance from the center. This concept is fundamental in understanding the strength and distribution of electric fields around conductive objects.

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