Step 1: Understanding the Concept:
Reactance is the opposition offered by an inductor or capacitor to the flow of alternating current.
For an inductor (a coil), the inductive reactance depends on the frequency of the applied voltage source.
Step 2: Key Formula or Approach:
The inductive reactance $X_L$ of a coil with inductance $L$ is given by:
\[ X_L = \omega L = 2\pi f L \]
where $\omega$ is the angular frequency and $f$ is the frequency of the source.
Step 3: Detailed Explanation:
Let's evaluate the reactance in both cases:
Case 1: Connected to an AC source.
An AC source has a non-zero frequency ($f_{\text{ac}}>0$).
Therefore, its inductive reactance will be a finite, non-zero positive value:
\[ X_{L(\text{ac})} = 2\pi f_{\text{ac}} L>0 \]
Case 2: Connected to a DC source.
A steady DC source has a constant voltage, meaning its frequency is zero ($f_{\text{dc}} = 0$).
Therefore, its inductive reactance is:
\[ X_{L(\text{dc})} = 2\pi (0) L = 0 \]
Now, calculate the required ratio of reactance when connected to AC versus DC:
\[ \text{Ratio} = \frac{X_{L(\text{ac})}}{X_{L(\text{dc})}} \]
\[ \text{Ratio} = \frac{2\pi f_{\text{ac}} L}{0} \]
Division by zero yields infinity ($\infty$). An ideal inductor offers no resistance to a steady DC current, only acting as a short circuit once the transient phase is over.
Step 4: Final Answer:
The ratio of its reactance is $\infty$.