Question

An LC circuit containing a 2.00-Hinductor oscillates at such a frequency that it radiates at 1.00-m a wavelength. (a) What is the capacitance of the circuit? (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?

Short answer

a. The capacitance of the circuit is$1.41\times {10}^{-19}\mathrm{F}$ .

b. The value of capacitance of the circuit is too small.

c. The assumption about the inductance of the circuit is inconsistent

Step-by-step solution

Definition of Concept

Wavelength: 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 capacitance of the circuit

(a)

Considering the given information,

The inductance of the circuit,L=2.00H

Wavelength,$\mathrm{\lambda }=1.00\mathrm{m}$

Apply the formula,

The LC circuit's frequency is stated as,

role="math" localid="1655963504576" $\mathrm{f}=\frac{1}{2\mathrm{\tau \tau }\sqrt{\mathrm{LC}}}......\left(1\right)$

Here,

- The circuit's inductance is L.

- The circuit's capacitance is C.

The following is the relationship between frequency and wavelength:

role="math" localid="1655963563368" $\mathrm{f}=\frac{\mathrm{c}}{\mathrm{I}}......\left(2\right)$

Here,

- The speed of light is c.

- The wavelength is .

Equation (1) and equation (2) must be solved.

$$\begin{gathered} \mathrm{C}=\frac{\left(1.00{\mathrm{m}}^2\right)}{4{\mathrm{\tau \tau }}^2\left(2.00\mathrm{H}\right)\left(3\times {10}^8\mathrm{m}/\mathrm{s}\right)} \\ =1.41\times {10}^{-19}\mathrm{F} \\ \mathrm{Therefore},\mathrm{the}\mathrm{required}\mathrm{capacitance}\mathrm{of}\mathrm{the}\mathrm{circuit}\mathrm{is}1.41\times {10}^{-19}\mathrm{F}. \end{gathered}$$

Find the unreasonable about this result

(b)

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

the small capacitance of the circuit is the irrational aspect of the outcome.

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

Find the assumptions are unreasonable or inconsistent

c)

The inductance value in the given LC circuit is far too high. As a result, the assumption

about the LC circuit's inductance is incorrect.

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