91影视

Capacitance of a parallel plate combination is given by the formula, $C=\frac{{蔚}_{0}A}{d}$, where C is the capacitance, ${蔚}_0$ is the permittivity of free space,A is the area of the plates, d is the distance between the plates.

Inductance of a solenoid is given by the formula $L=\frac{{渭}_0{N}^{2}A}{I}$, where, L is the inductance,N is the number of turns in the solenoid, A is the area of cross section of the solenoid, I is the length of the solenoid.

Resonant angular frequency of an A/C circuit is the angular frequency corresponding to which, the current in the circuit is maximum, or in other words the resistance is the circuit is minimum.The formula for resonant frequency is given by,$蝇=\frac{1}{\sqrt{LC}}$

We just discussed that, the capacitance is calculated by the formula,$C=\frac{{蔚}_{0}A}{d}$ . Given that,

$$\begin{gathered} {\mathrm{蔚}}_0=8.85脳{10}^{-12}{\mathrm{m}}^{-3}{\mathrm{kg}}^{-1}{\mathrm{s}}^4{\mathrm{A}}^2 \\ \mathrm{A}={\left(4.5脳{10}^{-2}\right)}^2{\mathrm{m}}^2 \\ \mathrm{d}=8脳{10}^{-3}\mathrm{m} \end{gathered}$$

Therefore that value of the capacitance is-

$$\begin{gathered} \mathrm{C}=\frac{8.85脳{10}^{-12}脳{\left(4.5脳{10}^{-2}\right)}^2}{8脳{10}^{-3}} \\ =22.4脳{10}^{-13}\mathrm{F} \end{gathered}$$

Similarly, for the inductor,$L=\frac{{渭}_0{N}^{2}A}{I}$ . The given parameters are-

$$\begin{gathered} {渭}_0=12.57脳{10}^{-7}H/m \\ N=125脳9 \\ =1125 \\ A=\mathrm{蟺}{\left(\frac{0.5}{\mathrm{A}}脳{10}^{-2}\right)}^2 \\ =1.96脳{10}^{-5}{\mathrm{m}}^2 \\ \mathrm{I}=9脳{10}^{-2}\mathrm{m} \end{gathered}$$

Therefore, the value of inductance is-

$$\begin{gathered} L=\frac{12.57脳{10}^{-7}脳{\left(1125\right)}^2脳1.96脳{10}^{-5}}{9脳{10}^{-2}} \\ =3.45脳{10}^{-4}H \end{gathered}$$

Hence the resonant frequency can be calculated using the formula as mentioned above-

$$\begin{gathered} \mathrm{蝇}=\frac{1}{\sqrt{\mathrm{LC}}} \\ =\frac{1}{\sqrt{3.45脳{10}^{-4}脳22.4脳{10}^{-13}}} \\ =3.6脳{10}^7\mathrm{rad}/\mathrm{s} \end{gathered}$$

Hence, the resonant angular frequency of the circuit is$3.6脳{10}^7\mathrm{rad}/\mathrm{s}$