To solve this problem, let us approach it step by step.

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Given Data:

1. Resistance of non-inductive resistor, $R_1 = 10 \, \Omega$
2. Resistance of the coil, $R_2 = 2 \, \Omega$
3. Total current, $I = 10 \, \text{A}$
4. Supply voltage, $V = 200 \, \text{V}$
5. Supply frequency, $f = 50 \, \text{Hz}$

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Step 1: Total Impedance ($Z$)

The total impedance is found using Ohm's law: $Z = \frac{V}{I} = \frac{200}{10} = 20 \, \Omega$

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Step 2: Reactance ($X_L$) and Inductance ($L$)

The impedance of the series combination is given by: $Z = \sqrt{(R_{\text{total}})^2 + X_L^2}$ where $R_{\text{total}} = R_1 + R_2 = 10 + 2 = 12 \, \Omega$.

Rearranging to find $X_L$: $X_L = \sqrt{Z^2 - R_{\text{total}}^2} = \sqrt{20^2 - 12^2} = \sqrt{400 - 144} = \sqrt{256} = 16 \, \Omega$

The inductive reactance $X_L$ is related to inductance $L$ by: $X_L = 2 \pi f L$ $L = \frac{X_L}{2 \pi f} = \frac{16}{2 \pi (50)} = \frac{16}{314.16} \approx 0.0509 \, \text{H}$

Thus, the inductance of the coil is: $L \approx 0.0509 \, \text{H}$

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Step 3: Power Factor

The power factor ($\cos \phi$) is given by: $\cos \phi = \frac{R_{\text{total}}}{Z} = \frac{12}{20} = 0.6$

Thus, the power factor is: $\cos \phi = 0.6$

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Step 4: Voltage Across the Coil

The coil's voltage has two components: resistive ($V_R$) and reactive ($V_L$): $V_R = I R_2 = 10 \times 2 = 20 \, \text{V}$ $V_L = I X_L = 10 \times 16 = 160 \, \text{V}$

The total voltage across the coil is: $V_{\text{coil}} = \sqrt{V_R^2 + V_L^2} = \sqrt{20^2 + 160^2} = \sqrt{400 + 25600} = \sqrt{26000} \approx 161.2 \, \text{V}$

Thus, the voltage across the coil is: $V_{\text{coil}} \approx 161.2 \, \text{V}$

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Final Answers:

1. Inductance of the coil: $L \approx 0.0509 \, \text{H}$
2. Power factor: $\cos \phi = 0.6$
3. Voltage across the coil: $V_{\text{coil}} \approx 161.2 \, \text{V}$

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Would you like further details or have any specific questions? 😊

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Related Questions:

1. How does the power factor affect the efficiency of the circuit?
2. Can we calculate the power consumed in this circuit? How?
3. What is the reactive power and apparent power in this circuit?
4. How would the values change if the frequency were doubled?
5. What happens to the voltage across the coil if the current increases?

Tip:

In AC circuits, always separate resistive and reactive components when analyzing voltage, current, and power.