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The carrier frequency generated by a tank circuit containing $1nF$ capacitor and $10\mu H$ inductor is(A) 1592 Hz(B) 1592 MHz(C) 1592 kHz(D) 159.2 Hz

Last updated date: 21st Apr 2024
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Hint: The given circuit is a Inductor-Capacitor circuit i.e. LC Circuit. It is also known that the source is alternating current. Use the resonant frequency formula to calculate the required carrier frequency.

Complete step by step solution:
The given circuit has an inductor of inductance $10\mu H$and a capacitance of$1nF$. We know that the frequency of AC circuits can be altered according to the requirements to pass and reject selective frequency for current. These are called resonant circuits.
The above given circuit is a LC resonant circuit. The frequency of the circuit is represented mathematically as
$\upsilon = \dfrac{1}{{2\pi \times \sqrt {L \times C} }}$
Where L is the Inductance value in $\mu H$ and C is capacitance value in Farad.
In order to understand the given terms in terms of single Farads and Henry, we expand the term
$1\mu H = {10^{ - 6}}H$ and $1nF = {10^{ - 9}}F$.
Substituting the values in the above frequency equation we get,
$\upsilon = \dfrac{1}{{2\pi \times \sqrt {L \times C} }}$
$\Rightarrow \upsilon = \dfrac{1}{{2\pi \times \sqrt {10 \times {{10}^{ - 6}}H \times 1 \times {{10}^{ - 9}}F} }}$
Adding up power values of 10 ,we get
$\Rightarrow \upsilon = \dfrac{1}{{2\pi \times \sqrt {10 \times {{10}^{ - 15}}} }}$
$\Rightarrow \upsilon = \dfrac{1}{{2\pi \times \sqrt {{{10}^{ - 14}}} }}$
Removing the square root value, we get
$\Rightarrow \upsilon = \dfrac{1}{{2\pi \times {{10}^{ - 7}}}}$
$\Rightarrow \upsilon = 0.1591 \times {10^7}$
Analyze all our options, we know that 1 kilohertz is equal to ${10^3}Hz$ and 1 MegaHertz is equal to ${10^6}Hz$. By using these, we can deduce the final frequency value.
Therefore
$\Rightarrow \upsilon = 1592 \times {10^3}Hz$
Which implies
$\upsilon = 1592kHz$

Therefore, Option (C) is the right answer.

Note
An LC tank circuit is a customized RLC circuit with zero resistance. It contains an inductor and a capacitor in either series or parallel connection. Tank circuits are frequently used as signal generators and band pass filters, which are used to select signals of particular frequency from clusters of signals. They are used in amplifiers, oscillators, and are a very commonly used circuit in electronics.