Question

The effective value of current is $ {\text{5A}} $ . Then the power dissipated is $ \left( {{\text{R = 10ohm, }}{{\text{X}}_{\text{L}}}{\text{ = 15ohm, }}{{\text{X}}_{\text{C}}}{\text{ = 25ohm}}} \right) $ .
(A) 150 $ {\text{W}} $
(B) 200 $ {\text{W}} $
(C) 250 $ {\text{W}} $
(D) $ 250\sqrt {\text{2}} {\text{W}} $

Answer 1

Hint: We only need to apply the impedance formula. Then by simply applying the power formula we can get the answer.

Formula used: As required here we need ohm’s law:
 $ {\text{Z = }}\sqrt {{{\text{R}}^{\text{2}}}{\text{ + }}{{\left( {{{\text{X}}_{\text{C}}}{\text{ - }}{{\text{X}}_{\text{L}}}} \right)}^{\text{2}}}} $
And
 $ {\text{P = }}{{\text{I}}^{\text{2}}}{\text{Z}} $
 $ {\text{Z}} $ is the Impedance,
 $ {\text{R}} $ is the resistance,
 $ {\text{I}} $ is the current .

Complete step by step answer:
We are already provided with the inductive reactance $ {{\text{X}}_{\text{L}}}{\text{ = 15\Omega }} $
And capacitive reactance $ {{\text{X}}_{\text{C}}}{\text{ = 25\Omega }} $
Now, $ {\text{Z = }}\sqrt {{{\text{R}}^{\text{2}}}{\text{ + }}{{\left( {{{\text{X}}_{\text{C}}}{\text{ - }}{{\text{X}}_{\text{L}}}} \right)}^{\text{2}}}} $
 $ {\text{Z = }}\sqrt {{\text{1}}{{\text{0}}^{\text{2}}}{\text{ + 1}}{{\text{0}}^{\text{2}}}} {\text{ = 10}}\sqrt {\text{2}} $
And the power dissipated is calculated:
 $ {\text{P = }}{{\text{I}}^{\text{2}}}{\text{Z}} $
 $ \Rightarrow {\text{P = }}{{\text{5}}^{\text{2}}}{\text{ \times 10}}\sqrt {\text{2}} {\text{ = 250}}\sqrt {\text{2}} {\text{W}} $
So, we need to see from the above options, and select the appropriate value.
Thus, the correct answer is option D.

Additional Information
Impedance is a measure of the overall opposition of a circuit to current, in other words it is how much the circuit impedes the flow of charge. It is like resistance, but it also takes into account the effects of capacitance and inductance. Impedance is measured in ohms. $ {{\text{X}}_{\text{L}}}{\text{, }}{{\text{X}}_{\text{C}}} $ are inductive and capacitive reactance. Reactant is the opposition to flow of current in an AC circuit and it is frequency dependent. It is equivalent to resistance in DC circuits where the inductor works as a normal conductor and the capacitor works as an open terminal.

Note:
There are two major types of circuit connection: Series and Parallel. We understood that a series circuit is one in which the current remains the same along with each element. An LCR circuit, also known as a resonant circuit, tuned circuit, or an RLC circuit, is an electrical circuit consisting of an inductor, capacitor and resistor connected in series or parallel. The LCR circuit analysis can be understood better in terms of phasors. A phasor is a rotating quantity. Also, there is no difference between an RLC circuit and an LCR circuit except for the order of the symbol represented in the circuit diagram.