Find the peak value and mean value of current over a half cycle if effective current in 40 Hz AC circuit is 7 A .

Answer

To solve the problem of finding the peak value and mean value of current over a half cycle in a 40 Hz AC circuit with an effective current (IRMS) of 7 A, we can follow these steps:

### Step 1: Understand the relationship between IRMS and peak current (I0)
The effective current (IRMS) is related to the peak current (I0) by the formula:
\[ I_{RMS} = \frac{I_0}{\sqrt{2}} \]
This can be rearranged to find the peak current:
\[ I_0 = I_{RMS} \times \sqrt{2} \]

### Step 2: Calculate the peak current (I0)
Given that \(I_{RMS} = 7 \, A\):
\[ I_0 = 7 \, A \times \sqrt{2} \]
Using the approximate value of \(\sqrt{2} \approx 1.414\):
\[ I_0 = 7 \, A \times 1.414 \approx 9.898 \, A \]
Thus, the peak current \(I_0\) is approximately:
\[ I_0 \approx 9.89 \, A \]

### Step 3: Calculate the mean current (Im) over a half cycle
The mean value of current over a half cycle is given by the formula:
\[ I_m = \frac{I_0}{\pi/2} = \frac{2I_0}{\pi} \]
Substituting the value of \(I_0\):
\[ I_m = \frac{2 \times 9.89 \, A}{\pi} \]
Using \(\pi \approx 3.14\):
\[ I_m = \frac{19.78 \, A}{3.14} \approx 6.29 \, A \]

### Final Results
- The peak value of current \(I_0\) is approximately **9.89 A**.
- The mean value of current \(I_m\) over a half cycle is approximately **6.29 A**.