Two alternating currents are given by `I_1=I_0 sin (omegat-phi)` and `I_2=I_1 cos (omega t+phi)` . What is the ratio of virtual values of the two currents?

Answer

To find the ratio of the virtual (RMS) values of the two alternating currents given by \( I_1 = I_0 \sin(\omega t - \phi) \) and \( I_2 = I_0 \cos(\omega t + \phi) \), we can follow these steps:

### Step 1: Understand the RMS Value Formula
The RMS (Root Mean Square) value of an alternating current \( I \) is given by the formula:
\[ I_{RMS} = \frac{I_0}{\sqrt{2}} \]
where \( I_0 \) is the maximum (peak) value of the current.

### Step 2: Calculate the RMS Value of \( I_1 \)
For the first current \( I_1 = I_0 \sin(\omega t - \phi) \):
- The maximum value \( I_{1, \text{max}} = I_0 \).
- Therefore, the RMS value is:
\[ I_{1, RMS} = \frac{I_{1, \text{max}}}{\sqrt{2}} = \frac{I_0}{\sqrt{2}} \]

### Step 3: Calculate the RMS Value of \( I_2 \)
For the second current \( I_2 = I_0 \cos(\omega t + \phi) \):
- The maximum value \( I_{2, \text{max}} = I_0 \).
- Therefore, the RMS value is:
\[ I_{2, RMS} = \frac{I_{2, \text{max}}}{\sqrt{2}} = \frac{I_0}{\sqrt{2}} \]

### Step 4: Find the Ratio of the RMS Values
Now, we can find the ratio of the RMS values of the two currents:
\[ \text{Ratio} = \frac{I_{1, RMS}}{I_{2, RMS}} = \frac{\frac{I_0}{\sqrt{2}}}{\frac{I_0}{\sqrt{2}}} \]
Since both RMS values are equal, we have:
\[ \text{Ratio} = 1 \]

### Final Result
The ratio of the virtual values (RMS values) of the two currents \( I_1 \) and \( I_2 \) is:
\[ \text{Ratio} = 1 : 1 \]