A coil having a resistance of `50.0 Omega` and an inductance of `0.5 H` is connected to an `AC` source of `110`, `v, 50Hz`.
The rms current in the circuit is

Options

A. `1.5A`
B. `0.67A`
C. `1A`
D. `3A`

Answer

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

### Step 1: Identify the given values
- Resistance (R) = 50.0 Ω
- Inductance (L) = 0.5 H
- RMS Voltage (V_RMS) = 110 V
- Frequency (f) = 50 Hz

### Step 2: Calculate the inductive reactance (X_L)
The inductive reactance (X_L) can be calculated using the formula:
\[ X_L = 2 \pi f L \]
Substituting the values:
\[ X_L = 2 \pi (50) (0.5) \]
\[ X_L = 2 \pi (25) \]
\[ X_L = 50 \pi \approx 157.08 \, \Omega \]

### Step 3: Calculate the impedance (Z)
The total impedance (Z) in an R-L circuit can be calculated using the formula:
\[ Z = \sqrt{R^2 + X_L^2} \]
Substituting the values:
\[ Z = \sqrt{(50)^2 + (157.08)^2} \]
\[ Z = \sqrt{2500 + 24649.61} \]
\[ Z = \sqrt{27149.61} \approx 164.69 \, \Omega \]

### Step 4: Calculate the RMS current (I_RMS)
The RMS current can be calculated using Ohm's law for AC circuits:
\[ I_{RMS} = \frac{V_{RMS}}{Z} \]
Substituting the values:
\[ I_{RMS} = \frac{110}{164.69} \approx 0.668 \, A \]

### Final Answer
The RMS current in the circuit is approximately **0.668 A**.

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