Consider two isolated, charged conducting spheres. One is a...

Consider two isolated, charged conducting spheres. One is a large sphere, and the second is smaller with a radius four times smaller than that of the large sphere but with four times as much charge. We denote the potential of the large sphere by $V_{L}$ and the electric field at its surface by $E_{L}$. The corresponding quantities for the smaller sphere are $V_{S}$ and $E_{S}$. a. Compute the ratio $E_{L} / E_{s}$. b. Find $V_{L} / V_{S}$

Consider two isolated, charged conducting spheres. One is a large sphere, and the second is smaller with a radius four times smaller than that of the large sphere but with four times as much charge. We denote the potential of the large sphere by $V_{L}$ and the electric field at its surface by $E_{L}$. The corresponding quantities for the smaller sphere are $V_{S}$ and $E_{S}$.
a. Compute the ratio $E_{L} / E_{s}$.
b. Find $V_{L} / V_{S}$

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Key Concepts

Electric Potential of a Sphere

The electric potential on the surface of a sphere is defined by V = kQ/R. This linear dependence on charge and inverse dependence on radius help in understanding how the potential varies when either the charge or the size of the sphere changes.

Scaling Laws in Electrostatics

Scaling laws describe how physical quantities like electric field and potential change when dimensions or charge values are scaled. They are essential for predicting the effects of altering the radius or charge on a sphere, as seen in problems comparing spheres of different sizes and charge magnitudes.

Conducting Sphere

A conducting sphere has its charge uniformly distributed along its surface, which simplifies the analysis of its electric field and potential. This concept is fundamental when studying isolated charged conductors because symmetry allows us to apply Coulomb's law directly to determine field and potential at the surface.

Electric Field at the Surface

The electric field at the surface of a charged sphere is given by E = kQ/R², where Q is the charge and R is the radius. This relation shows the inverse square dependency on the radius, making it a crucial concept when comparing spheres of different sizes or charges.

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