An ideal solenoid with length \(\ell\), cross-sectional area \(A\), and \(n\) turns per unit length has negligible internal resistance and is connected in series with an open switch, a resistor of resistance \(R\), and a battery of emf \(\mathcal{E}\). A second solenoid has the same length and cross-sectional area, but has \(2n\) turns per unit length and negligible internal resistance. If the second solenoid replaces the first in the circuit and the switch is closed until steady state is reached in each case, what is the ratio of the initial maximum magnetic energy stored in the solenoid to the new maximum magnetic energy stored, \(\dfrac{U_1}{U_2}\)?