To solve this problem, we need to calculate a few values related to the RLC circuit consisting of a resistor (coil), inductor, and capacitor connected in series.
Given Data
- Resistance \( R = 25 , \Omega \)
- Inductance \( L = 0.19 , H \)
- Capacitance \( C = 125 \times 10^{-6} , F \)
- Voltage \( V = 200 , V \)
- Frequency \( f = 50 , Hz \)
Formulas
- Angular frequency \( \omega \): \[ \omega = 2\pi f \]
- Inductive reactance \( X_L \): \[ X_L = \omega L \]
- Capacitive reactance \( X_C \): \[ X_C = \frac{1}{\omega C} \]
- Impedance \( Z \): \[ Z = \sqrt{R^2 + (X_L - X_C)^2} \]
- Total current \( I \): \[ I = \frac{V}{Z} \]
- Voltage across the coil \( V_R \): \[ V_R = I \cdot R \]
- Voltage across the capacitor \( V_C \): \[ V_C = I \cdot X_C \]
Step-by-Step Calculation
A) Impedance Calculation
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Calculate \( \omega \): \[ \omega = 2 \pi (50) \approx 314.16 , rad/s \]
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Calculate \( X_L \): \[ X_L = \omega L = 314.16 \times 0.19 \approx 59.68 , \Omega \]
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Calculate \( X_C \): \[ X_C = \frac{1}{\omega C} = \frac{1}{314.16 \times 125 \times 10^{-6}} \approx 25.37 , \Omega \]
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Calculate \( Z \): \[ Z = \sqrt{R^2 + (X_L - X_C)^2} = \sqrt{(25)^2 + (59.68 - 25.37)^2} \] \[ Z = \sqrt{625 + (34.31)^2} = \sqrt{625 + 1175.80} \approx \sqrt{1800.80} \approx 42.43 , \Omega \]
B) Total Current Calculation
\[ I = \frac{V}{Z} = \frac{200}{42.43} \approx 4.71 , A \]
C) Voltage Across the Coil Calculation
\[ V_R = I \cdot R = 4.71 \cdot 25 \approx 117.75 , V \]
D) Voltage Across the Capacitor Calculation
\[ V_C = I \cdot X_C = 4.71 \cdot 25.37 \approx 119.44 , V \]
Summary of Results
- A) Impedance \( Z \approx 42.43 , \Omega \)
- B) Total current \( I \approx 4.71 , A \)
- C) Voltage across the coil \( V_R \approx 117.75 , V \)
- D) Voltage across the capacitor \( V_C \approx 119.44 , V \)
These calculations confirm the behavior of the RLC series circuit under the given conditions.
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