The current in the series L-C-R circuit is

Options

A. `i=i_(m)sin(omegat+phi)`
B. `i=(v_(m))/(sqrt(R^(2)+(X_(c)-X_(L))^(2)))sin(omegat+phi)`
C. `i=2i_(m)cos(omegat+phi)`
D. Both (a) and (b)

Explanation

To find the expression for the current in a series L-C-R circuit, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Circuit**:
We have a series L-C-R circuit where an alternating current (AC) voltage \( V(t) = V_0 \sin(\omega t) \) is applied. The circuit consists of a resistor (R), an inductor (L), and a capacitor (C) connected in series.

2. **Identify the Voltage Drops**:
The voltage across each component can be expressed as:
- Voltage across the resistor: \( V_R = I \cdot R \)
- Voltage across the inductor: \( V_L = I \cdot j\omega L \) (where \( j \) is the imaginary unit)
- Voltage across the capacitor: \( V_C = \frac{I}{j\omega C} \)

3. **Calculate the Impedance (Z)**:
The total impedance \( Z \) of the circuit can be calculated using the formula:
\[ Z = \sqrt{R^2 + (X_L - X_C)^2} \]
where \( X_L = \omega L \) (inductive reactance) and \( X_C = \frac{1}{\omega C} \) (capacitive reactance). Therefore, the impedance becomes:
\[ Z = \sqrt{R^2 + \left(\omega L - \frac{1}{\omega C}\right)^2} \]

4. **Determine the Current**:
The current \( I(t) \) in the circuit can be expressed as:
\[ I(t) = I_m \sin(\omega t + \phi_0) \]
where \( I_m \) is the maximum current and \( \phi_0 \) is the phase angle. The maximum current can be calculated using Ohm's law for AC circuits:
\[ I_m = \frac{V_m}{Z} \]
where \( V_m \) is the peak voltage.

5. **Express the Current**:
Substituting for \( I_m \):
\[ I(t) = \frac{V_m}{Z} \sin(\omega t + \phi_0) \]

6. **Phase Angle**:
The phase angle \( \phi_0 \) depends on the relationship between the inductive and capacitive reactances:
\[ \tan(\phi_0) = \frac{X_L - X_C}{R} \]
This indicates whether the circuit is inductive or capacitive.

### Final Expression:
Thus, the expression for the current in the series L-C-R circuit is:
\[ I(t) = \frac{V_m}{Z} \sin(\omega t + \phi_0) \]