A metal sphere carrying an evenly distributed charge will...

A metal sphere carrying an evenly distributed charge will have spherical equipotential surfaces surrounding it. Suppose the sphere's radius is 50.0 $\mathrm{cm}$ and it carries a total charge of $+1.50 \mu \mathrm{C}$ (a) Calculate the potential of the sphere's surface.(b) You want to draw equipotential surfaces at intervals of 500 $\mathrm{V}$ outside the sphere's surface. Calculate the distance between the first and the second equipotential surfaces, and between the 20 $\mathrm{th}$ and 21 $\mathrm{st}$ equipotential surfaces. (c) What does the changing spacing of the surfaces tell you about the electric field?

Key Concepts

Relationship Between Electric Field and Potential Gradient

The electric field is related to the spatial rate of change (gradient) of the electric potential, expressed by the equation E = -dV/dr for radial fields. This connection shows that a small change in distance can correspond to a large change in potential when the field is strong, indicating that closer to the charge, where equipotential surfaces are more closely spaced, the electric field is stronger, and further away, the surfaces spread out as the electric field weakens.

Nonlinear Spatial Variation of Potential

Due to the inverse relationship between the electric potential and the distance from the charge, equal changes in potential do not correspond to equal increments in radial distance. This nonlinearity implies that the spacing between equipotential surfaces varies with distance from the charge, offering insight into how the electric field strength diminishes with increasing distance.

Electric Potential of a Charged Sphere

This concept explains how to compute the electric potential at a point due to a spherically symmetric charge distribution. For a charged conducting sphere, its external electric potential is the same as if the entire charge were concentrated at its center. The potential at any point outside the sphere is given by the formula V = kQ/r, where Q is the total charge, r is the distance from the center of the sphere, and k is Coulomb's constant.

Equipotential Surfaces

Equipotential surfaces are locations in space where the electric potential is constant. For a spherically symmetric charge distribution, these surfaces form concentric spheres centered at the charge. They are important because no work is done when moving a charge along an equipotential surface, and their spacing is directly related to the strength of the electric field.

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