Question

In a pure inductive circuit, a sinusoidal voltage \(V(t) = 200 \sin 250t\) is applied to a pure inductance of \(L = 0.02\,\text{H}\). The current through the coil is:

• A. \(40 \sin \left(250t - \frac{\pi}{2}\right)\)
• B. \(40 \cos \left(250t - \frac{\pi}{2}\right)\)
• C. \(40 \sin \left(250t + \frac{\pi}{2}\right)\)
• D. \(40 \cos \left(250t + \frac{\pi}{2}\right)\)

Hint:

In a pure inductor, current always lags voltage by \(90^\circ\), and \(V_0 = \omega L I_0\).

Answer 1

The Correct Option is A

Step 1: Recall relation for pure inductor.
In a pure inductive circuit, current lags voltage by \(\frac{\pi}{2}\).
\[ i(t) = I_0 \sin\left(\omega t - \frac{\pi}{2}\right) \]

Step 2: Identify given quantities.
\[ V(t) = V_0 \sin \omega t \]
So,
\[ V_0 = 200,\quad \omega = 250\,\text{rad/s} \]

Step 3: Use relation between voltage and current amplitude.
For inductor:
\[ V_0 = \omega L I_0 \]

Step 4: Substitute values.
\[ 200 = 250 \times 0.02 \times I_0 \]
\[ 200 = 5 I_0 \]
\[ I_0 = 40 \]

Step 5: Write expression for current.
\[ i(t) = 40 \sin\left(250t - \frac{\pi}{2}\right) \]

Step 6: Interpretation.
Current lags voltage by \(90^\circ\), which is characteristic of pure inductive circuits.

Step 7: Final answer.
\[ \boxed{40 \sin \left(250t - \frac{\pi}{2}\right)} \]