Match the following
`{:(,"Currents","r.m.s. values"),((1),x_(0) sin omega t,(i)" "x),((2),x_(0) sin omega t cos omega t,(ii)" "x_(0)/sqrt(2)),((3),x_(0) sin omega t+x_(0) cos omega t,(iii)" "x_(0)/((2sqrt(2)))):}`

Options

A. 1.(i),2.(ii),3.(iii)
B. 1.(ii),2.(iii),3.(i)
C. 1.(i),2.(iii),3.(ii)
D. none of these

Answer

To solve the problem of matching the currents with their respective r.m.s. values, we will follow these steps:

### Step 1: Identify the first current and calculate its r.m.s. value.
- **Current:** \( x_0 \sin(\omega t) \)
- **Peak Value:** \( x_0 \)
- **RMS Value Formula:**
\[ \text{RMS} = \frac{\text{Peak Value}}{\sqrt{2}} = \frac{x_0}{\sqrt{2}} \]
- **Match:** This corresponds to option (ii) \( \frac{x_0}{\sqrt{2}} \).

### Step 2: Identify the second current and calculate its r.m.s. value.
- **Current:** \( x_0 \sin(\omega t) \cos(\omega t) \)
- **Using the identity:**
\[ \sin(\omega t) \cos(\omega t) = \frac{1}{2} \sin(2\omega t) \]
- **Peak Value:**
\[ \text{Peak Value} = \frac{x_0}{2} \]
- **RMS Value:**
\[ \text{RMS} = \frac{\text{Peak Value}}{\sqrt{2}} = \frac{x_0/2}{\sqrt{2}} = \frac{x_0}{2\sqrt{2}} \]
- **Match:** This corresponds to option (iii) \( \frac{x_0}{2\sqrt{2}} \).

### Step 3: Identify the third current and calculate its r.m.s. value.
- **Current:** \( x_0 \sin(\omega t) + x_0 \cos(\omega t) \)
- **RMS Calculation:**
- The two terms have a phase difference of \( \frac{\pi}{2} \).
- RMS value is calculated as:
\[ \text{RMS} = \sqrt{ \left( \frac{x_0}{\sqrt{2}} \right)^2 + \left( \frac{x_0}{\sqrt{2}} \right)^2 } \]
\[ = \sqrt{ \frac{x_0^2}{2} + \frac{x_0^2}{2} } = \sqrt{x_0^2} = x_0 \]
- **Match:** This corresponds to option (i) \( x_0 \).

### Final Matching:
1. \( x_0 \sin(\omega t) \) matches with \( \frac{x_0}{\sqrt{2}} \) (ii)
2. \( x_0 \sin(\omega t) \cos(\omega t) \) matches with \( \frac{x_0}{2\sqrt{2}} \) (iii)
3. \( x_0 \sin(\omega t) + x_0 \cos(\omega t) \) matches with \( x_0 \) (i)

### Summary of Matches:
- (1) \( x_0 \sin(\omega t) \) → (ii) \( \frac{x_0}{\sqrt{2}} \)
- (2) \( x_0 \sin(\omega t) \cos(\omega t) \) → (iii) \( \frac{x_0}{2\sqrt{2}} \)
- (3) \( x_0 \sin(\omega t) + x_0 \cos(\omega t) \) → (i) \( x_0 \)