Question

A RLC circuit is in its resonance condition. Its circuit components have value \[R = 50\Omega \] , \[L = 2H\] , \[C = 0.5mF\], \[V = 250V\]. Then find the power in the circuit.
A. \[6kW\]
B. \[10kW\]
C. \[12kW\]
D. \[12.5kW\]

Answer 1

Hint:First of all, we need to understand the concept of the RLC Circuit before proceeding with the question. Basically, in an RLC circuit the Resistance \[R\] , Inductor \[L\] and capacitor \[C\] are connected in series to each other in a circuit. In an RLC circuit, it is basically an oscillating circuit with a resistor, inductor, and capacitor connected in series. At first, Capacitor is transferred by a charge, and by this charge flows through the capacitor, voltage causes current to flow towards the inductor to basically discharge the capacitor in a circuit.

Formula used:
As it is mentioned in the question, the circuit is in resonance, Therefore we have the equation of Power as:
\[P = i_{rms}^2 \times R\]
\[P = \dfrac{{{V^2}}}{R}\]

Complete step by step solution:
 As we know for resonance circuit conditions,
\[{X_l} = {X_c}\]
Since, Impedance \[Z = R\] ,
\[i_{rms}^2 = \,\dfrac{V}{Z}\] ,
Now, substituting values on the equation above we get,
\[P = i_{rms}^2 \times R\]
\[\Rightarrow P = \dfrac{{{V^2}}}{R} = \dfrac{{250 \times 250}}{5}\]
\[\Rightarrow P = 12500\dfrac{J}{s}\]
which is equal to \[P = 12.5kW\]

Hence, option D i.e. \[12.5kW\] is the correct answer

Note: For the Resonance circuit, \[{X_l} = {X_c}\]and with the help of this condition we can calculate the power transmitted. Resonance circuits have a very low impedance and are built by using an inductor for example coil, that is connected parallel to a capacitor. The resonance in the RLC circuit usually occurs when inductive and capacitive reactance are equal in their magnitude but cancel each other out since they are 180 degrees apart in their phase.