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To solve this problem, let’s break down the calculation steps based on the information provided.

### Given Data:
1. A three-phase heater with three resistors, each of 20 ohms, connected in a star (Y) configuration.
2. The line voltage (\( V_L \)) is 220 V.

### Calculations:

#### Step 1: Calculate the Phase Voltage (\( V_f \)) for a Star (Y) Configuration
In a star configuration, the phase voltage \( V_f \) is related to the line voltage \( V_L \) by the equation:
\[
V_f = \frac{V_L}{\sqrt{3}}
\]
Substituting \( V_L = 220 \, \text{V} \):
\[
V_f = \frac{220}{\sqrt{3}} \approx 127.02 \, \text{V}
\]

#### Step 2: Calculate the Phase Current (\( I_f \))
The phase current \( I_f \) is given by Ohm's law:
\[
I_f = \frac{V_f}{R}
\]
where \( R = 20 \, \Omega \).
\[
I_f = \frac{127.02}{20} \approx 6.35 \, \text{A}
\]

#### Step 3: Calculate the Line Current (\( I_L \))
For a star-connected load, the line current \( I_L \) is equal to the phase current \( I_f \):
\[
I_L = I_f = 6.35 \, \text{A}
\]

#### Step 4: Calculate the Total Active Power (\( P \))
The total active power in a three-phase system is given by:
\[
P = 3 \times V_f \times I_f = \sqrt{3} \times V_L \times I_L
\]
Using the second formula:
\[
P = \sqrt{3} \times 220 \times 6.35 \approx 2420 \, \text{W} = 2.42 \, \text{kW}
\]

### Answer
The correct answer is:
- **Option (a)**: \( I_L = 6.35 \, \text{A} \) and \( P = 2.42 \, \text{kW} \).

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Would you like further details on any part of this solution? Here are some related questions that could deepen understanding:

1. What would change in the calculations if the configuration was delta instead of star?
2. How does the line current differ between star and delta configurations?
3. How would we calculate total power if the resistances were not equal?
4. What is the impact of line voltage increase on power consumption in a star-connected load?
5. How can reactive and apparent power be calculated in this three-phase system?

**Tip:** In three-phase systems, knowing the type of connection (star or delta) is crucial, as it directly impacts voltage, current, and power calculations.

Um aquecedor trifásico é constituído de três resistência de 20Ω ligadas em estrela. Calcular a corrente de linha e a potência ativa total, sendo a tensão de linha 220V.

Calculate the line current and total active power of a three-phase heater with three 20Ω resistors connected in a star configuration and a line voltage of 220V. This solution covers three-phase power systems, star configuration calculations, and Ohm's law applications.

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