When two charged conductors are connected by a wire, charge flows between them until they acquire the same electric potential ($V_1 = V_2$) [Source 53].

1. Potential Condition: For spheres of radius $a$ and $b$ with charges $Q_a$ and $Q_b$: $V_a = V_b \Rightarrow \frac{kQ_a}{a} = \frac{kQ_b}{b} \Rightarrow \frac{Q_a}{Q_b} = \frac{a}{b}$ [Source 41].

2. Electric Field Ratio: The electric field at the surface of a conducting sphere is given by $E = \frac{kQ}{R^2}$ [Source 41]. The ratio of the fields is: $\frac{E_a}{E_b} = \frac{kQ_a / a^2}{kQ_b / b^2} = \left( \frac{Q_a}{Q_b} \right) \times \left( \frac{b^2}{a^2} \right)$

3. Substitution: Substituting the charge ratio derived in step 1: $\frac{E_a}{E_b} = \left( \frac{a}{b} \right) \times \left( \frac{b^2}{a^2} \right) = \frac{b}{a}$