The equation of alternating current for a circuit is given `I = 50 cos 100 pi t`. Find (i) frequency of a.c. applied (ii) mean value of current during positive half of the cycle (iii) virtual value of current and (iv) the value of current 1/300 sec after it was zero

Answer

To solve the given problem step by step, we will break down each part of the question regarding the alternating current equation \( I = 50 \cos(100 \pi t) \).

### Step 1: Find the frequency of the AC applied
The general equation for alternating current is given by:
\[ I = I_0 \cos(2 \pi f t) \]
where \( I_0 \) is the maximum current and \( f \) is the frequency.

From the given equation:
\[ I = 50 \cos(100 \pi t) \]
we can compare it with the general equation. Here, we have:
\[ 2 \pi f = 100 \pi \]
To find the frequency \( f \), we divide both sides by \( 2 \pi \):
\[ f = \frac{100 \pi}{2 \pi} = 50 \text{ Hz} \]

### Step 2: Find the mean value of current during the positive half of the cycle
The mean value \( I_{\text{mean}} \) of an AC current during the positive half cycle can be calculated using the formula:
\[ I_{\text{mean}} = 0.637 I_0 \]
Substituting \( I_0 = 50 \):
\[ I_{\text{mean}} = 0.637 \times 50 = 31.85 \text{ A} \]

### Step 3: Find the virtual value (RMS value) of current
The root mean square (RMS) value \( I_{\text{rms}} \) is given by:
\[ I_{\text{rms}} = \frac{I_0}{\sqrt{2}} \]
Substituting \( I_0 = 50 \):
\[ I_{\text{rms}} = \frac{50}{\sqrt{2}} \approx 35.35 \text{ A} \]

### Step 4: Find the value of current \( \frac{1}{300} \) sec after it was zero
To find the current at \( t = \frac{1}{300} \) seconds, we substitute \( t \) into the equation:
\[ I = 50 \cos(100 \pi t) \]
Substituting \( t = \frac{1}{300} \):
\[ I = 50 \cos\left(100 \pi \times \frac{1}{300}\right) = 50 \cos\left(\frac{100 \pi}{300}\right) = 50 \cos\left(\frac{\pi}{3}\right) \]
Since \( \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \):
\[ I = 50 \times \frac{1}{2} = 25 \text{ A} \]

### Final Answers:
1. Frequency of AC applied: **50 Hz**
2. Mean value of current during positive half of the cycle: **31.85 A**
3. Virtual value (RMS value) of current: **35.35 A**
4. Value of current \( \frac{1}{300} \) sec after it was zero: **25 A**