The ratio of peak value and r.m.s value of an alternating current is

Options

A. 1
B. `1/2`
C. `sqrt(2)`
D. `1//sqrt(2)`

Answer

To find the ratio of the peak value (I_max) to the root mean square (r.m.s) value (I_rms) of an alternating current, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Definitions**:
- The peak value (I_max) of an alternating current is the maximum value of the current.
- The r.m.s value (I_rms) is a measure of the effective value of the alternating current.

2. **Write the Equation for Alternating Current**:
- The general equation for an alternating current can be expressed as:
\[ I(t) = I_0 \sin(\omega t) \]
- Here, \(I_0\) is the amplitude (or peak value) of the current.

3. **Identify the Peak Value**:
- From the equation, we can identify that the peak value \(I_{\text{max}} = I_0\).

4. **Calculate the r.m.s Value**:
- The r.m.s value for an alternating current is given by the formula:
\[ I_{\text{rms}} = \frac{I_{\text{max}}}{\sqrt{2}} = \frac{I_0}{\sqrt{2}} \]

5. **Find the Ratio of Peak Value to r.m.s Value**:
- Now, we need to find the ratio of the peak value to the r.m.s value:
\[ \text{Ratio} = \frac{I_{\text{max}}}{I_{\text{rms}}} \]
- Substituting the values we have:
\[ \text{Ratio} = \frac{I_0}{\frac{I_0}{\sqrt{2}}} \]

6. **Simplify the Expression**:
- Simplifying the ratio:
\[ \text{Ratio} = \frac{I_0 \cdot \sqrt{2}}{I_0} = \sqrt{2} \]

7. **Conclusion**:
- Therefore, the ratio of the peak value to the r.m.s value of an alternating current is:
\[ \text{Ratio} = \sqrt{2} \]

### Final Answer:
The ratio of peak value to r.m.s value of an alternating current is \(\sqrt{2}\).

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