Question

a metal sphere of radius r has an electric charge +q on it. part a determine an expression for the electric potential v on the spheres surface. express your answer in terms of some or all of the variables r, q. use k for the constant from coulombs law. part b use the definition of capacitance to determine an expression for the capacitance of a metal sphere of radius r. (hint: assume the other plate is infinitely far away.) express your answer in terms of some or all of the variables r, q. use k for the constant from coulombs law.

Explanation:

Step1: Recall electric - potential formula

The electric potential \(V\) at the surface of a charged metal sphere of radius \(R\) with charge \(q\) is given by the formula \(V = k\frac{q}{R}\), where \(k\) is the Coulomb's law constant.

Step2: Recall capacitance formula

The capacitance \(C\) is defined as \(C=\frac{q}{V}\). Substitute \(V = k\frac{q}{R}\) into the capacitance formula. Then \(C=\frac{q}{k\frac{q}{R}}\).

Step3: Simplify the capacitance expression

Simplifying \(C=\frac{q}{k\frac{q}{R}}\), the \(q\) terms cancel out, and we get \(C=\frac{R}{k}\).

Answer:

Part A: \(V = k\frac{q}{R}\)
Part B: \(C=\frac{R}{k}\)