Question

An LC circuit containing a $\mathbf{1}\mathbf{.}\mathbf{00}\mathbf{-}\mathbf{pF}$capacitor oscillates at such a frequency that it radiates at a \({\rm{300 - nm}}\)wavelength. (a) What is the inductance of the circuit? (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?

Short answer

.(a) The inductance of the circuit is$2.53\times {10}^{-20}\mathrm{H}$ .

(b) The value of the inductance of the circuit is too small.

(c) The assumption about the wavelength of the circuit is inconsistent

Step-by-step solution

Definition of Concept

The distance between identical points (adjacent crests) in adjacent cycles of a waveform signal propagated in space or along a wire is defined as the wavelength. This length is typically specified in wireless systems in meters (m), centimeters (cm), or millimeters (mm).

Find the inductance of the circuit

(a)

Considering the given information,

The capacitance of the circuit,$\mathrm{C}=1.00\mathrm{pF}\left(\frac{{10}^{-12}\mathrm{F}}{1\mathrm{pF}}\right)=1.00\times {10}^{-12}\mathrm{F}$

Wavelength,$\lambda =300\mathrm{nm}\left(\frac{{10}^{-9}\mathrm{m}}{1\mathrm{nm}}\right)=3.00\times {10}^{-7}\mathrm{m}$

Apply the formula,

The frequency of the circuit is calculated as follows:

$\mathrm{f}=\frac{1}{2\mathrm{\pi }\sqrt{\mathrm{LC}}}$……………..(1)

Here,

The circuit's inductance is L and the circuit's capacitance is C.

The relationship between frequency and wavelength can be expressed as,

$\mathrm{f}=\frac{\mathrm{c}}{\mathrm{\lambda }}$………………….(2)

Here, the speed of light is c, and the wavelength is .

Equation (1) and equation (2) must be solved to get,

$\mathrm{L}=\frac{{\mathrm{\lambda }}^2}{\left(4{\mathrm{\pi }}^2\right)\times \mathrm{C}\times \left({\mathrm{C}}^2\right)}$

Substituting the given values in the above expression, we get,

$$\begin{gathered} \mathrm{L}=\frac{{\left(3.00\times {10}^{-7}\mathrm{m}\right)}^2}{4\mathrm{\pi }\left(1.00\times {10}^{-12}\mathrm{F}\right){\left(3\times {10}^8\mathrm{m}/\mathrm{s}\right)}^2} \\ =2.53\times {10}^{-20}\mathrm{H} \end{gathered}$$

$$\begin{gathered} \mathrm{L}=\frac{{\left(3.00\times {10}^{-7}\mathrm{m}\right)}^2}{4{\mathrm{\pi }}^2\left(1.00\times {10}^{-12}\mathrm{F}\right){\left(3\times {10}^8\mathrm{m}/\mathrm{s}\right)}^2} \\ =2.53\times {10}^{-20}\mathrm{H} \end{gathered}$$

Therefore, the required inductance of the circuit is $2.53\times {10}^{-20}\mathrm{H}$.

Find the unreasonable about this result

(b)

The LC circuit's predicted inductance is far too little to be useful in practice. As a result,

the modest inductance of the circuit is the irrational aspect of the outcome.

Therefore, the value of the inductance of the circuit is too small.

Find the assumptions are unreasonable or inconsistent

(c)

The wavelength value provided is insufficient.

As a result, the wavelength assumption is unreliable.

Therefore, the assumption about the wavelength of the circuit is inconsistent.