To solve this problem, we need to analyze the series circuit consisting of a resistor and an inductor connected to an AC source. We will calculate the current flowing through the circuit, the voltage across each component, and then check if the sum of these voltages exceeds the source voltage.
1. Calculating the Current in the Circuit
The first step is to determine the total impedance of the circuit. The impedance (Z) in a series R-L circuit can be calculated using the formula:
Z = √(R² + (XL)²)
Where:
- R is the resistance (31.4 ohms)
- XL is the inductive reactance, calculated as XL = 2πfL
- f is the frequency (50 Hz)
- L is the inductance (0.1 H)
First, we calculate the inductive reactance:
XL = 2π × 50 × 0.1 = 31.4 ohms
Now, we can find the total impedance:
Z = √(31.4² + 31.4²) = √(986.76 + 986.76) = √(1973.52) ≈ 44.4 ohms
Next, we can calculate the current (I) using Ohm's Law:
I = V / Z
Substituting the values:
I = 200V / 44.4 ohms ≈ 4.5 A
2. Voltage Across the Inductor and Resistor
Now that we have the current, we can find the voltage across the resistor (VR) and the inductor (VL).
The voltage across the resistor is given by:
VR = I × R
Substituting the values:
VR = 4.5 A × 31.4 ohms ≈ 141.3 V
Next, we calculate the voltage across the inductor:
VL = I × XL
Substituting the values:
VL = 4.5 A × 31.4 ohms ≈ 141.3 V
3. Comparing the Voltages
Now, let's check if the algebraic sum of the voltages across the inductor and resistor is greater than the source voltage:
Sum of Voltages = VR + VL = 141.3 V + 141.3 V = 282.6 V
Since the source voltage is 200 V, we can see that:
282.6 V > 200 V
This means that the sum of the voltages across the inductor and resistor indeed exceeds the source voltage. This is a characteristic of AC circuits, where the voltages can be out of phase, leading to a situation where their algebraic sum is greater than the source voltage.
Summary of Results
- Current in the circuit: Approximately 4.5 A
- Voltage across the resistor: Approximately 141.3 V
- Voltage across the inductor: Approximately 141.3 V
- Sum of voltages across inductor and resistor: 282.6 V (greater than source voltage)
This analysis illustrates the behavior of R-L circuits in AC systems, highlighting the importance of impedance and phase relationships in determining circuit characteristics.
