If the alternating current I`=I_(1)cosomegat+I_(2) sinomegat`, then the rms current is given by

Options

A. `(I_(1)+I_(2))/(sqrt(2))`
B. `(|I_(1)+I_(2)|)/(sqrt(2))`
C. `sqrt((I_(1)^(2)+I_(2)^(2))/2)`
D. `sqrt((I_(1)^(2)+I_(2)^(2))/sqrt(2)`

Answer

To find the RMS (Root Mean Square) current for the given alternating current \( I = I_1 \cos(\omega t) + I_2 \sin(\omega t) \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Components**:
The alternating current is given as:
\[ I = I_1 \cos(\omega t) + I_2 \sin(\omega t) \]
Here, \( I_1 \) and \( I_2 \) are the amplitudes of the cosine and sine components, respectively.

2. **Express the Current in Terms of a Single Amplitude**:
We can express \( I \) in terms of a single amplitude \( I \) using the Pythagorean identity:
\[ I^2 = I_1^2 + I_2^2 \]
This follows from the fact that \( \sin^2(\theta) + \cos^2(\theta) = 1 \).

3. **Calculate the RMS Value**:
The RMS value of a sinusoidal current is given by:
\[ I_{\text{rms}} = \frac{I}{\sqrt{2}} \]
Substituting \( I \) from the previous step, we have:
\[ I_{\text{rms}} = \frac{\sqrt{I_1^2 + I_2^2}}{\sqrt{2}} \]

4. **Final Expression for RMS Current**:
Therefore, the expression for the RMS current can be simplified to:
\[ I_{\text{rms}} = \frac{\sqrt{I_1^2 + I_2^2}}{\sqrt{2}} = \sqrt{\frac{I_1^2 + I_2^2}{2}} \]

5. **Conclusion**:
The correct option for the RMS current is:
\[ I_{\text{rms}} = \sqrt{\frac{I_1^2 + I_2^2}{2}} \]