For series RLC circuit given below in figure, choose the correct answer based on Kirchoff's voltage law from following:
\(Ri+L\frac{di}{dt}+\frac{1}{C}\int i\,dt = V(t)\)
To solve the problem using Kirchoff's Voltage Law (KVL) for a series RLC circuit, we need to consider the voltage drops across the resistor, inductor, and capacitor, as well as the input voltage source \( V(t) \).
- For the resistor \( R \), the voltage drop is given by Ohm's Law as \( Ri \), where \( i \) is the current flowing through the circuit.
- For the inductor \( L \), the voltage drop is given by \( L \frac{di}{dt} \), where \(\frac{di}{dt}\) is the rate of change of current.
- For the capacitor \( C \), the voltage drop is related to the charge \( q \) on the capacitor by \(\frac{q}{C} = \frac{1}{C}\int i \, dt\), since the current \( i \) is the rate of change of charge \( q \), i.e., \( i = \frac{dq}{dt} \).
According to KVL, the sum of the voltages around a closed loop must be equal to the applied voltage \( V(t) \). Therefore, the equation for the circuit becomes:
\(Ri + L \frac{di}{dt} + \frac{1}{C} \int i \, dt = V(t)\)
This corresponds to the second option given in the list.
Conclusion: The correct answer is \(Ri + L \frac{di}{dt} + \frac{1}{C} \int i \, dt = V(t)\), as it accurately represents the application of Kirchoff's Voltage Law to the series RLC circuit.
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