Step 1: Understanding the Concept:
In a series LR alternating current circuit, the voltage leads the current by a certain phase angle $\phi$.
This phase angle is determined by the ratio of the inductive reactance ($X_L$) to the resistance ($R$).
Step 2: Key Formula or Approach:
The formula for the phase angle $\phi$ is:
\[ \tan\phi = \frac{X_L}{R} \]
where $X_L = 2\pi f L$ is the inductive reactance.
$f$ is the frequency of the AC source, and $L$ is the inductance.
Step 3: Detailed Explanation:
Given values:
Resistance, $R = 200\ \Omega$
Inductance, $L = \frac{1}{2\pi}\text{ H}$
Frequency, $f = 100\text{ Hz}$
First, calculate the inductive reactance $X_L$:
\[ X_L = 2\pi f L \]
\[ X_L = 2\pi \times 100 \times \left( \frac{1}{2\pi} \right) \]
The $2\pi$ terms cancel out:
\[ X_L = 100\ \Omega \]
Now, calculate the tangent of the phase angle:
\[ \tan\phi = \frac{X_L}{R} \]
\[ \tan\phi = \frac{100}{200} \]
\[ \tan\phi = \frac{1}{2} = 0.5 \]
Therefore, the phase angle is:
\[ \phi = \tan^{-1}(0.5) \]
Step 4: Final Answer:
The phase angle is $\tan^{-1}(0.5)$.