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Question: An inductance coil of \({{1H}}\) and a condenser of capacity \({{1pF}}\) produce resonance. The reso...

An inductance coil of 1H{{1H}} and a condenser of capacity 1pF{{1pF}} produce resonance. The resonant frequency will be:
A) 106πHz\dfrac{{{{1}}{{{0}}^{{6}}}}}{{{\pi }}}{{Hz}}
B) 27π×166Hz{{27\pi \times 1}}{{{6}}^{{6}}}{{Hz}}
C) 2π106Hz\dfrac{{{{2\pi }}}}{{{{1}}{{{0}}^{{6}}}}}{{Hz}}
D) 1062πHz\dfrac{{{{1}}{{{0}}^{{6}}}}}{{{{2\pi }}}}{{Hz}}

Explanation

Solution

We know that the formula of resonant frequency, by substituting the values, we can solve the above question. Electrical resonance occurs in an AC circuit when the two reactances which are opposite and equal cancel each other and the point on the graph at which this happens is where the two reactance curves cross each other.

Formula used:
F(resonant frequency) = 12πLC\dfrac{{{1}}}{{{{2\pi }}\sqrt {{{LC}}} }}
Where L is inductance & C is capacitance
& 1pF=10−12Farad{{1}}{{pF = 1}}{{{0}}^{{{ - 12}}}}{{Farad}}

Complete step by step answer:
Given that, the inductance of a coil is 1Hq{{1}}{{{H}}_{{q}}} and the capacity of a condenser is equal to C=1pF{{C = 1pF}} , now converting the capacitance in farad we get
C=1pF=10−12F{{C = 1pF}}{{ = }}{{1}}{{{0}}^{{{ - 12}}}}{{F}} So, C=10−12F{{C = }}{{1}}{{{0}}^{{{ - 12}}}}{{F}}
Now,
Putting the value of L&C; in formula f=12πLC{{f = }}\dfrac{{{1}}}{{{{2\pi }}\sqrt {{{LC}}} }} we get
f=12π1×10−12{{f}}{{ = }}\dfrac{{{1}}}{{{{2\pi }}\sqrt {{{1 \times 1}}{{{0}}^{{{ - 12}}}}} }}

⇒f=12π10−12 \Rightarrow {{f}}{{ = }}\dfrac{{{1}}}{{{{2\pi }}\sqrt {{{1}}{{{0}}^{{{ - 12}}}}} }}
∴f=1062πHz\therefore {{f}}{{ = }}\dfrac{{{{1}}{{{0}}^{{6}}}}}{{{{2\pi }}}}{{{H}}_{{z}}}

So, the correct option is (D) i.e. f=1062πHz{{f}}{{ = }}\dfrac{{{{1}}{{{0}}^{{6}}}}}{{{{2\pi }}}}{{{H}}_{{z}}}.

Note: Resonance is an important concept in oscillatory motion. The resonant frequency is the characteristic frequency of a body or a system that reaches the maximum degree of oscillations.
In an electrical system, the resonant frequency is defined as the frequency at which the transfer function reaches its maximum value. This for a given input, the maximum output can be obtained. It has been proud that the resonance is obtained when the capacitive impedance and the inductive impedance values are equal. In this article, we will discuss the resonant frequency formula with examples. The resonant circuits are used to create a particular frequency or to select a particular frequency form a complex circuit. So, the resonant frequency f=12πLC.{{f}}{{ = }}\dfrac{{{1}}}{{{{2\pi }}\sqrt {{{LC}}} }}.