Two metal spheres of different sizes are charged such that...

Two metal spheres of different sizes are charged such that the electric potential is the same at the surface of each. Sphere $A$ has a radius three times that of sphere $B$. Let $Q_A$ and $Q_B$ be the charges on the two spheres, and let $E_A$ and $E_B$ be the electric-field magnitudes at the surfaces of the two spheres. What are (a) the ratio $Q_B/Q_A$ and (b) the ratio $E_B/E_A$?

Key Concepts

Ratio Analysis in Electrostatics

This concept involves the use of proportional reasoning to compare different physical quantities, like charge and electric field, between systems with varying geometries. By analyzing ratios and applying the fundamental formulas for potential and field, one can extract relationships that are independent of the constant factors, thereby simplifying complex comparisons in electrostatic problems.

Electric Potential of a Conducting Sphere

This concept centers around the formula V = kQ/R, which shows that the electric potential at the surface of a charged sphere depends directly on its charge Q and inversely on its radius R. It is a fundamental principle in electrostatics that allows one to understand how variations in size or charge affect the potential experienced at the surface of a conductor.

Electric Field at the Surface of a Sphere

Derived from Gauss's law, the electric field on the surface of a charged sphere is given by E = kQ/R². This relationship is essential for explaining how the distribution of charge on a spherical conductor results in a specific field intensity at its surface, highlighting the dependencies between charge, radius, and field strength.

Relationship Between Charge and Radius for Equal Potentials

This key concept focuses on the proportionality that arises when two spheres are maintained at the same electric potential. According to V = kQ/R, if the potentials are equal, the charge on each sphere must be directly proportional to its radius. This principle is crucial when comparing systems where geometry impacts how much charge is required to achieve a given potential.

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