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

Name: Problem 34, Chapter 22: Essential University Physics
Uploaded: 2020-04-27T14:24:31-08:00
Duration: 8 min 36 s
Channel: Manish Haldankar
Description: A large metal sphere has three times the diameter of a smaller sphere and carries three 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 three times the diameter of a smaller sphere and carries three 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 three times the diameter of a smaller sphere and carries three 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.

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Key Concepts

Scaling relationships in electrostatics

Scaling relationships refer to how physical quantities change with size. In the context of charged spheres, if the radius is scaled by a factor and the charge by another factor, the potential scales with the ratio Q/R while the electric field scales with Q/R². This analysis is crucial in comparing the effects of simultaneous variations in size and charge on the potential and the electric field.

Electric potential of a charged sphere

For an isolated charged conducting sphere, the potential at its surface is given by V = kQ/R, where k is Coulomb’s constant, Q is the total charge, and R is the radius of the sphere. This relation implies that for a larger sphere with a proportionally increased charge, the potential can remain the same because the decrease in potential due to a larger radius may be offset by an increase in the charge.

Electric field at the surface of a charged sphere

The electric field at the surface of a uniformly charged conducting sphere is given by E = kQ/R². This expression shows that, for a given charge, the electric field decreases with the square of the radius. Thus, if a sphere is larger even with an increased charge, the electric field at its surface can be smaller compared to that of a smaller sphere.

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