Question

The peak value of an alternating emf $e$ given by $e = e_0 \cos \omega t$ is $10\text{ V}$ and its frequency is $50\text{ Hz}$. At time $t = \frac{1}{600}\text{ s}$, the instantaneous e.m.f is $\left[\cos \frac{\pi}{6} = \sin \frac{\pi}{3} = \frac{\sqrt{3}}{2}\right]$

• A. $10\text{ V}$
• B. $\frac{1}{\sqrt{3}}\text{ V}$
• C. $5\text{ V}$
• D. $5\sqrt{3}\text{ V}$

Hint:

When evaluating alternating current phase arguments with a frequency of $50\text{ Hz}$, the term $2\pi f$ becomes exactly $100\pi$. Multiplying this instantly by your given time component ($1/600$) simplifies directly to a standard reference angle fraction ($\pi/6$), saving valuable test time.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
An alternating electromotive force (e.m.f.) is defined by the sinusoidal function $e = e_0 \cos \omega t$.
We are given the peak voltage amplitude ($e_0 = 10\text{ V}$) and the linear cyclic frequency ($f = 50\text{ Hz}$). We need to compute the exact instantaneous value of this voltage at the specific time mark $t = \frac{1}{600}\text{ s}$.

Step 2: Key Formula or Approach:
1. The relationship between angular frequency $\omega$ and linear frequency $f$ is given by:
$$\omega = 2\pi f$$ 2. Substituting this into the instantaneous e.m.f. equation yields:
$$e = e_0 \cos(2\pi f t)$$

Step 3: Detailed Explanation:
First, let's calculate the value of the phase angle argument ($\theta = 2\pi f t$) using our given values ($f = 50\text{ Hz}$ and $t = \frac{1}{600}\text{ s}$):
$$\theta = 2\pi \times 50 \times \frac{1}{600}$$ $$\theta = 100\pi \times \frac{1}{600} = \frac{100\pi}{600} = \frac{\pi}{6}\text{ radians}$$ Now, substitute this evaluated phase angle and the peak voltage ($e_0 = 10\text{ V}$) back into the primary e.m.f. equation:
$$e = 10 \cos\left(\frac{\pi}{6}\right)$$ The problem statement provides the standard trigonometric constant value $\cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}$. Substitute this into the equation:
$$e = 10 \times \frac{\sqrt{3}}{2} = 5\sqrt{3}\text{ V}$$

Step 4: Final Answer:
The instantaneous e.m.f at that moment is $5\sqrt{3}\text{ V}$, which corresponds to option (D).