Question

Assertion (A): In a series R-L-C circuit the voltage across resistor, inductor and capacitor are $8V,16V$ and $10V$ respectively. The resultant emf in the circuit is $10V$ .
Reason (R): Resultant emf of the circuit is given by the relation, $E = \sqrt {V_R^2 + {{({V_L} - {V_C})}^2}} $ .
A. Both assertion and reason are correct and reason is the correct explanation for assertion.
B. Both assertion and reason are correct but reason is not the correct explanation for the answer.
C. Assertion is correct but reason is not correct.
D. Both assertion and reason are incorrect.

Answer 1

Hint: In this question first we will find the net emf of the circuit and then we will check the condition. The emf formula is used to calculate the net emf of the LCR series circuit; the emf of the LCR series circuit is dependent on the voltage across the inductor. The voltage across the capacitor and the voltage across the resistance are two different voltages. The net emf can be calculated using this method.

Formula used:
Emf of a L-C-R series circuit is given by the formula:
$E = \sqrt {V_R^2 + {{({V_L} - {V_C})}^2}} $
Where, the emf of the circuit, $E$, the voltage of the inductor, ${V_L}$, the voltage of the capacitor, ${V_C}$ and the voltage of the resistance, ${V_R}$ .

Complete answer:
According to the question,
The voltage across the inductor is, ${V_L} = 16$
The voltage across the capacitor is, ${V_C} = 10$
The voltage across the resistance is, ${V_R} = 8$
And we know that the resultant emf in the R-L-C circuit is given by:
$E = \sqrt {V_R^2 + {{({V_L} - {V_C})}^2}} $ -----(1)
Now, substituting the voltage across the inductor, voltage across the capacitor and the voltage across the resistance in the above equation (1), then the above equation (1) is written as,
$E = \sqrt {{8^2} + {{(16 - 10)}^2}} \\
\Rightarrow E = \sqrt {64 + 36} \\
\Rightarrow E = \sqrt {100} \\
\therefore E = 10\,V $
Thus, the above equation shows the net emf of the LCR series circuit which is $10V$. Hence, from this we can say that both assertion as well as reason are correct and reason is the correct explanation for assertion.

Therefore, the correct option is A.

Note: The square root of the sum of the squares of the voltage across resistance and the square of the difference between the voltage across the inductor and the voltage across the capacitor is the emf of the LCR series circuit.