Consider a circuit consisting of a capacitor of capacitance $$C$$ and a coil with $$N$$ turns per unit length, cross sectional area $$S$$ and length $$d$$, where $$d^2\gg S$$. There is another coil of length $$d/2$$, cross sectional area $$S/2$$ and $$2N$$ turns per unit length completely inside the larger coil, as shown in the figure. The ends of this smaller coil are connected with each other by an insulated conducting wire. The self-inductance of the larger coil is $$L$$. Neglecting edge effects and all the Ohmic resistances, the resonant frequency of the circuit is: