An LCR series circuit is under resonance. If `I_(m)` is current amplitude, `V_(m)` is voltage amplitude, R is the resistance, Z is the impedance, `X_(L)` is the inductive reactance and `X_(C )` is the capacitive reactance, then

Options

A. `I_(m)=(Z)/(V_(m))`
B. `I_(m)=(V_(m))/(X_(L))`
C. `I_(m)=(V_(m))/(X_(C ))`
D. `I_(m)=(V_(m))/(R )`

Explanation

To solve the problem regarding the LCR series circuit under resonance, we can follow these steps:

### Step 1: Understand Resonance in LCR Circuit
In an LCR series circuit, resonance occurs when the inductive reactance \(X_L\) is equal to the capacitive reactance \(X_C\). This means:
\[ X_L = X_C \]

### Step 2: Impedance at Resonance
At resonance, the impedance \(Z\) of the circuit is minimized. The general formula for impedance in an LCR circuit is given by:
\[ Z = \sqrt{R^2 + (X_L - X_C)^2} \]
Since \(X_L = X_C\) at resonance, the term \((X_L - X_C)\) becomes zero. Thus, the impedance simplifies to:
\[ Z = R \]

### Step 3: Current Amplitude Equation
The current amplitude \(I_m\) in an LCR circuit can be expressed using the voltage amplitude \(V_m\) and the impedance \(Z\):
\[ I_m = \frac{V_m}{Z} \]
Substituting \(Z\) with \(R\) (since \(Z = R\) at resonance), we get:
\[ I_m = \frac{V_m}{R} \]

### Conclusion
Thus, the equation for the current amplitude \(I_m\) in an LCR series circuit under resonance is:
\[ I_m = \frac{V_m}{R} \]