Question

The instantaneous value of an alternating current is given by $i = 50 \sin (100 \pi t)$. It will achieve a value of $25\text{ A}$ after a time interval of ($\sin 30^\circ = 0.5$)

• A. $\frac{1}{300}\text{ s}$
• B. $\frac{1}{100}\text{ s}$
• C. $\frac{1}{200}\text{ s}$
• D. $\frac{1}{600}\text{ s}$

Hint:

Whenever the target value is exactly half of the peak amplitude ($I = \frac{1}{2}I_0$) for a standard sine wave starting from zero, the phase angle is always $\frac{\pi}{6}$. Equating $\omega t = \frac{\pi}{6}$ lets you solve for time instantly without writing out any trigonometric lines!

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
The problem provides the instantaneous alternating current equation as a function of time, $i(t) = 50 \sin(100 \pi t)$. We need to determine the earliest time $t$ at which the current reaches an instantaneous magnitude of $25\text{ A}$.

Step 2: Key Formula or Approach:
We substitute the targeted instantaneous current value directly into the given harmonic AC wave function and solve the resulting trigonometric expression for the time variable $t$: $$i = I_0 \sin(\omega t)$$

Step 3: Detailed Explanation:
Set up the algebraic equality equation by plugging in $i = 25\text{ A}$: $$25 = 50 \sin(100 \pi t)$$ Isolate the sine term by dividing both sides by the peak amplitude factor of $50$: $$\sin(100 \pi t) = \frac{25}{50} = 0.5$$ We are given that $\sin 30^\circ = 0.5$. To match the arguments properly, convert the angle from degrees to circular radian units: $$30^\circ = 30 \times \frac{\pi}{180} = \frac{\pi}{6}\text{ radians}$$ Now equate the corresponding angular arguments of the sine functions: $$100 \pi t = \frac{\pi}{6}$$ Cancel out the transcendental constant $\pi$ from both sides of the expression: $$100 t = \frac{1}{6}$$ Isolate the time variable $t$ by dividing through by $100$: $$t = \frac{1}{600}\text{ s}$$

Step 4: Final Answer:
The current achieves a value of $25\text{ A}$ after a time interval of $\frac{1}{600}\text{ s}$, which perfectly matches option (D).