To calculate the inductance of the coil, we first need to determine the impedance and then the inductive reactance. Given: Resistance, $R = 30\, \Omega$ RMS Voltage, $V_{rms} = 100\, \text{V}$ RMS Current, $I_{rms} = 2\, \text{A}$ Step 1: Calculate the impedance ($Z$) of the coil. The impedance is the total opposition to current flow in an AC circuit and can be found using Ohm's law for AC circuits: $$Z = \frac{V_{rms}}{I_{rms}}$$ $$Z = \frac{100\, \text{V}}{2\, \text{A}}$$ $$Z = 50\, \Omega$$ Step 2: Calculate the inductive reactance ($X_L$). For a coil with resistance, the impedance is related to the resistance and inductive reactance by the formula: $$Z = \sqrt{R^2 + X_L^2}$$ We can rearrange this to solve for $X_L$: $$X_L^2 = Z^2 - R^2$$ $$X_L = \sqrt{Z^2 - R^2}$$ Substitute the values of $Z$ and $R$: $$X_L = \sqrt{(50\, \Omega)^2 - (30\, \Omega)^2}$$ $$X_L = \sqrt{2500\, \Omega^2 - 900\, \Omega^2}$$ $$X_L = \sqrt{1600\, \Omega^2}$$ $$X_L = 40\, \Omega$$ Step 3: Calculate the inductance ($L$). The inductive reactance is also related to the inductance ($L$) and the angular frequency ($\omega$) by the formula: $$X_L = \omega L$$ Where $\omega = 2\pi f$, and $f$ is the frequency of the AC source. The problem does not provide the frequency of the AC source. Therefore, we will express the inductance in terms of the angular frequency $\omega$. $$L = \frac{X_L}{\omega}$$ $$L = \frac{40\, \Omega}{\omega}$$ If we assume a standard AC frequency, for example, $f = 50\, \text{Hz}$ (a common frequency in many regions): The angular frequency would be: $$\omega = 2\pi f = 2\pi (50\, \text{Hz}) = 100\pi\, \text{rad/s}$$ Now, substitute this value into the equation for $L$: $$L = \frac{40\, \Omega}{100\pi\, \text{rad/s}}$$ $$L = \frac{2}{5\pi}\, \text{H}$$ $$L \approx \frac{2}{5 \times 3.14159}$$ $$L \approx \frac{2}{15.70795}$$ $$L \approx 0.1273\, \text{H}$$ The inductance of the coil is $\boxed{\frac{40}{\omega}\, \text{H}}$. Assuming a frequency of $50\, \text{Hz}$, the inductance is approximately $\boxed{0.127\, \text{H}}$.