Answer
Verified
400.2k+ views
Hint: Power factor is the cosine of the angle between net voltage and the current. At resonance, the magnitudes of ${{V}_{L}}$ and ${{V}_{c}}$ are equal. Since they are in opposite directions, their resultant will be zero. Hence the net voltage vector will be equal to ${{V}_{R}}$. And the angle between ${{V}_{R}}$ and i is zero.
Complete step by step answer:
In an AC circuit, the voltage (potential difference) across the components in the circuit and the current flowing in the circuit are treated as vectors. The phase difference between the current and the voltage across the respective component is the angle between the current vector and the voltage vector of the component.
Since the voltage across the resistance and the current are always in phase, the vectors of voltage of resistance (${{V}_{R}}$) and current (i) are parallel.
There is a phase difference of $+\dfrac{\pi }{2}$ between the inductor voltage and the current. That is the inductor voltage leads the current by a phase of $\dfrac{\pi }{2}$.
Therefore, the vector of inductor voltage (${{V}_{L}}$) is at an angle of $+\dfrac{\pi }{2}$ from the current vector.
There is a phase difference of $-\dfrac{\pi }{2}$ between the capacitor voltage and the current. That is the capacitor voltage lags behind the current by a phase of $\dfrac{\pi }{2}$.
Therefore, the vector of capacitor voltage (${{V}_{c}}$) is at an angle of $-\dfrac{\pi }{2}$ from the current vector.
The entire can be summarised with the help of the diagram given below.
Power factor is equal to $\cos \alpha $. From the diagram we get that
At resonance, the magnitudes of ${{V}_{L}}$ and ${{V}_{c}}$ are equal. Since they are in opposite directions, their resultant will be zero. Hence the net voltage vector will be equal to ${{V}_{R}}$. And the angle between ${{V}_{R}}$ and i is zero.
Hence, $\alpha =0$, which means that $\cos \alpha =\cos 0=1$ .
Therefore, the value of the power factor for the L-C-R series circuit at resonance is equal to one or unity.
Hence, the correct option is C.
Note: Power factor is defined as the ratio of the power consumed by the circuit (resistance) to the power supplied to by the source of the circuit. In other words, is the ratio of the power of heat generated to the power supplied.
Since at resonance the net voltage is only due to the resistance of the circuit, all the power will be consumed by the resistance. Hence, all the energy will be converted into heat and the value of power factor will be maximum.
Therefore, the value of the power factor is one.
Complete step by step answer:
In an AC circuit, the voltage (potential difference) across the components in the circuit and the current flowing in the circuit are treated as vectors. The phase difference between the current and the voltage across the respective component is the angle between the current vector and the voltage vector of the component.
Since the voltage across the resistance and the current are always in phase, the vectors of voltage of resistance (${{V}_{R}}$) and current (i) are parallel.
There is a phase difference of $+\dfrac{\pi }{2}$ between the inductor voltage and the current. That is the inductor voltage leads the current by a phase of $\dfrac{\pi }{2}$.
Therefore, the vector of inductor voltage (${{V}_{L}}$) is at an angle of $+\dfrac{\pi }{2}$ from the current vector.
There is a phase difference of $-\dfrac{\pi }{2}$ between the capacitor voltage and the current. That is the capacitor voltage lags behind the current by a phase of $\dfrac{\pi }{2}$.
Therefore, the vector of capacitor voltage (${{V}_{c}}$) is at an angle of $-\dfrac{\pi }{2}$ from the current vector.
The entire can be summarised with the help of the diagram given below.
Power factor is equal to $\cos \alpha $. From the diagram we get that
At resonance, the magnitudes of ${{V}_{L}}$ and ${{V}_{c}}$ are equal. Since they are in opposite directions, their resultant will be zero. Hence the net voltage vector will be equal to ${{V}_{R}}$. And the angle between ${{V}_{R}}$ and i is zero.
Hence, $\alpha =0$, which means that $\cos \alpha =\cos 0=1$ .
Therefore, the value of the power factor for the L-C-R series circuit at resonance is equal to one or unity.
Hence, the correct option is C.
Note: Power factor is defined as the ratio of the power consumed by the circuit (resistance) to the power supplied to by the source of the circuit. In other words, is the ratio of the power of heat generated to the power supplied.
Since at resonance the net voltage is only due to the resistance of the circuit, all the power will be consumed by the resistance. Hence, all the energy will be converted into heat and the value of power factor will be maximum.
Therefore, the value of the power factor is one.
Recently Updated Pages
Basicity of sulphurous acid and sulphuric acid are
Assertion The resistivity of a semiconductor increases class 13 physics CBSE
Three beakers labelled as A B and C each containing 25 mL of water were taken A small amount of NaOH anhydrous CuSO4 and NaCl were added to the beakers A B and C respectively It was observed that there was an increase in the temperature of the solutions contained in beakers A and B whereas in case of beaker C the temperature of the solution falls Which one of the following statements isarecorrect i In beakers A and B exothermic process has occurred ii In beakers A and B endothermic process has occurred iii In beaker C exothermic process has occurred iv In beaker C endothermic process has occurred
The branch of science which deals with nature and natural class 10 physics CBSE
What is the stopping potential when the metal with class 12 physics JEE_Main
The momentum of a photon is 2 times 10 16gm cmsec Its class 12 physics JEE_Main
Trending doubts
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
Write an application to the principal requesting five class 10 english CBSE
Difference Between Plant Cell and Animal Cell
a Tabulate the differences in the characteristics of class 12 chemistry CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
What organs are located on the left side of your body class 11 biology CBSE
Discuss what these phrases mean to you A a yellow wood class 9 english CBSE
List some examples of Rabi and Kharif crops class 8 biology CBSE