Step 1: Understanding the Concept:
In an alternating current (AC) series LCR circuit, the total opposition to the current flow is called impedance ($Z$). The amplitude of the current (peak current $I_0$) can be determined by dividing the peak voltage ($V_0$) by the total impedance of the circuit.
Step 2: Key Formula or Approach:
The formula for the impedance $Z$ in a series LCR circuit is:
\[ Z = \sqrt{R^2 + (X_L - X_C)^2} \]
The amplitude of the current is given by Ohm's law for AC circuits:
\[ I_0 = \frac{V_0}{Z} \]
Step 3: Detailed Explanation:
Given values:
Resistance, $R = 80 \, \Omega$
Inductive reactance, $X_L = 100 \, \Omega$
Capacitive reactance, $X_C = 40 \, \Omega$
Input voltage, $V(t) = 2500 \cos(100\pi t)$ V
From the voltage equation, the peak voltage is $V_0 = 2500$ V.
First, calculate the net reactance:
\[ X_L - X_C = 100 \, \Omega - 40 \, \Omega = 60 \, \Omega \]
Now, calculate the impedance $Z$:
\[ Z = \sqrt{80^2 + 60^2} \]
\[ Z = \sqrt{6400 + 3600} \]
\[ Z = \sqrt{10000} = 100 \, \Omega \]
Finally, calculate the peak current $I_0$:
\[ I_0 = \frac{2500}{100} = 25 \, \text{A} \]
Step 4: Final Answer:
The amplitude of the current in the circuit is 25 A. The correct option is (A).