Resonance Frequency

$$\omega = \frac{1}{\sqrt{\mathrm{L}_{\mathrm{eq}} \mathrm{C}}}$$

Now for $$\mathrm{L}_{\mathrm{eq}}$$, assume mutual inductance is M

Let current in outer coil is $$i_1$$ and induced current in inner coil is $$i_2$$

$$V_{\text{outer}} = L_1 \frac{d i_1}{d t} + M \frac{d i_2}{d t}$$

for inner coil, $$0 = L_2 \frac{d i_2}{d t} + M \frac{d i_1}{d t}$$

$$V_{\text{outer}} = \left(L_1 - \frac{M^2}{L_2}\right) \frac{d i_1}{d t}$$

$$L_{\text{eq.}} = L_1 - \frac{M^2}{L_2}$$

Now, $$L_1 = \mu_0 n^2 s d = L$$

$$L_2 = \mu_0 (2 n)^2 \frac{s}{2} \cdot \frac{d}{2} = L$$

for $$M \rightarrow$$ $$\phi_{\text{inner}} = (\mu_0 n i) \frac{s}{2} (2 n) \frac{d}{2}$$

$$\frac{\phi}{i} = M = \frac{L}{2}$$

Now, $$\omega = \frac{1}{\sqrt{\mathrm{L}_{\text{eq}} \cdot \mathrm{C}}}$$

$$\omega = \frac{2}{\sqrt{3 \mathrm{LC}}}$$