Question

In an oscillating LC circuit, the total stored energy is U and the maximum charge on the capacitor is Q. When the charge on the capacitor is \[\dfrac{Q}{2}\], the energy stored in the inductor is
\[\left( {\text{A}} \right){\text{ }}U/2\]
$\left( {\text{B}} \right){\text{ }}U/4$
\[\left( {\text{C}} \right){\text{ }}\left( {4/3} \right)U\]
\[\left( {\text{D}} \right){\text{ }}3U/4\]

Answer 1

Hint: The ability to store electrical charge in a capacitor is called electrical capacitance.
Capacitor: The two-terminal device that performs electrical capacitance is called a capacitor.
The capacitance of a capacitor varies with the size and shape of the capacitor, nature of the surrounding medium.
The presence of other capacitors in its neighborhood also affects the capacitance of the capacitor.
The total energy of the capacitor: The work done in charging the capacitor is stored in the form of electrical potential energy.
Inductor: It is a two-terminal device that stores electrical energy in a magnetic field when electricity flows through it.
Energy stored in the inductor: The energy stored in the inductor in presence of a magnetic field.

Formula used:
\[{\text{Q = CV}}\], here Q= Charge stored on either capacitor, C= Capacitance of the capacitor, V= Potential difference of the capacitor,
${\text{U = }}\dfrac{1}{2}\dfrac{{{{\text{Q}}^2}}}{{\text{C}}}$, here U= total energy of the capacitor.

Complete step by step answer:
It is given that Charge stored in the capacitor=Q/2
By using the formula and substituting the values we get, the total energy stored in the capacitor, ${\text{U = }}\dfrac{1}{2}\dfrac{{{{\text{Q}}^2}}}{{\text{C}}}$
Replacing Q by Q/2 we get, ${{\text{U}}_{\text{C}}} = \dfrac{1}{2}\dfrac{{{{{\text{(Q/2)}}}^2}}}{{\text{C}}}$
$=\dfrac{1}{2}\dfrac{{{{\text{Q}}^2}}}{{4{\text{C}}}}$
Exchanging the value we get,
$=\dfrac{1}{4}\left( {\dfrac{{{{\text{Q}}^2}}}{{{\text{2C}}}}} \right)$
Comparing U and UC we get, ${{\text{U}}_{\text{C}}} = \dfrac{1}{4}{\text{U}}$
The energy stored in the inductor is, ${{\text{U}}_{\text{L}}} = {\text{U - }}{{\text{U}}_{\text{C}}}$
$={\text{U - }}\dfrac{1}{4}{\text{U}}$
Taking LCM we get,
$=\dfrac{3}{4}{\text{U}}$

Hence, the correct option is \[\left( D \right)\].

Note:
The inductor is used to control signals; reactive magnetic fluxes are produced in the coil to counteract the changes in the electric current.
LC circuit is called a resonant circuit, tank circuit or tuned circuit, it comprises a capacitor and inductor.
Oscillating LC oscillator is used for producing high-frequency signals; LC oscillator is mainly used in radio, television.