The root mean square value of the alternating current is equal to

Options

A. Twice the peak value
B. Half the peak value
C. `1/(sqrt(2))` times the peak value
D. Equal to the peak value

Answer

To find the root mean square (RMS) value of an alternating current (AC), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding RMS Value**:
The RMS value of an alternating current is defined as the square root of the average of the squares of the instantaneous values over one complete cycle. Mathematically, it can be expressed as:
\[ I_{rms} = \sqrt{\frac{1}{T} \int_0^T I(t)^2 dt} \]
where \( T \) is the time period of the AC signal and \( I(t) \) is the instantaneous current.

2. **Expressing AC Current**:
For a sinusoidal AC current, we can express the instantaneous current as:
\[ I(t) = I_0 \sin(\omega t) \]
where \( I_0 \) is the peak current and \( \omega \) is the angular frequency.

3. **Calculating the Square**:
Now, we square the instantaneous current:
\[ I(t)^2 = I_0^2 \sin^2(\omega t) \]

4. **Using Trigonometric Identity**:
We can use the trigonometric identity:
\[ \sin^2(\theta) = \frac{1 - \cos(2\theta)}{2} \]
to rewrite the squared current:
\[ I(t)^2 = I_0^2 \cdot \frac{1 - \cos(2\omega t)}{2} \]

5. **Integrating Over One Complete Cycle**:
Now we need to integrate \( I(t)^2 \) over one complete cycle (from \( 0 \) to \( T \)):
\[ \int_0^T I(t)^2 dt = \int_0^T I_0^2 \cdot \frac{1 - \cos(2\omega t)}{2} dt \]
This can be split into two parts:
\[ = \frac{I_0^2}{2} \int_0^T dt - \frac{I_0^2}{2} \int_0^T \cos(2\omega t) dt \]

6. **Evaluating the Integrals**:
The first integral evaluates to:
\[ \int_0^T dt = T \]
The second integral evaluates to zero over one complete cycle because the integral of cosine over its period is zero:
\[ \int_0^T \cos(2\omega t) dt = 0 \]

7. **Final Calculation**:
Thus, we have:
\[ \int_0^T I(t)^2 dt = \frac{I_0^2}{2} T \]
Now, substituting this back into the RMS formula:
\[ I_{rms} = \sqrt{\frac{1}{T} \cdot \frac{I_0^2}{2} T} = \sqrt{\frac{I_0^2}{2}} = \frac{I_0}{\sqrt{2}} \]

8. **Conclusion**:
Therefore, the root mean square value of the alternating current is:
\[ I_{rms} = \frac{I_0}{\sqrt{2}} \]
This means the RMS value is \( \frac{1}{\sqrt{2}} \) times the peak value.

### Answer:
The root mean square value of the alternating current is \( \frac{I_0}{\sqrt{2}} \), which corresponds to option C.