A resistance of 40 `Omega` is connected in series with inductor of self-inductance 5 H and a capacitor of capacitance 80 `mu`F. This combination is connected to an AC source of rms voltage 220 V. frequency of AC source can changed continuously.
What rms current flows in circuit in a state of resonance ?

Options

A. 11 amp
B. 5.5 amp
C. 11`sqrt(2)` amp
D. `5.5 sqrt(2)` amp

Explanation

To find the RMS current flowing in the circuit at resonance, we can follow these steps:

### Step 1: Understand the Components in the Circuit
We have:
- Resistance (R) = 40 Ω
- Inductance (L) = 5 H
- Capacitance (C) = 80 μF = 80 × 10^-6 F
- RMS Voltage (V_RMS) = 220 V

### Step 2: Determine the Resonance Condition
In an LCR series circuit, resonance occurs when the inductive reactance (X_L) is equal to the capacitive reactance (X_C).

The formulas for reactance are:
- Inductive Reactance: \( X_L = \omega L \)
- Capacitive Reactance: \( X_C = \frac{1}{\omega C} \)

At resonance:
\[ X_L = X_C \]
\[ \omega L = \frac{1}{\omega C} \]

### Step 3: Calculate the Impedance at Resonance
At resonance, the impedance (Z) of the circuit is simply equal to the resistance (R):
\[ Z = R \]

### Step 4: Calculate the RMS Current
The RMS current (I_RMS) can be calculated using Ohm's law for AC circuits:
\[ I_{RMS} = \frac{V_{RMS}}{Z} \]

Since at resonance \( Z = R \):
\[ I_{RMS} = \frac{V_{RMS}}{R} \]

### Step 5: Substitute the Values
Substituting the known values:
\[ I_{RMS} = \frac{220 \, V}{40 \, \Omega} \]

### Step 6: Perform the Calculation
\[ I_{RMS} = 5.5 \, A \]

### Final Answer
The RMS current flowing in the circuit at resonance is **5.5 A**.

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