In an a.c. circuit, the instantaneous e.m.f. and current are given by e = 100 sin 30 t
`i = 20 sin( 30t-(pi)/(4))` In one cycle of a.c., the average power consumed by the circuit and the wattless current are, respectively :

Options

A. 50,0
B. 50,10
C. `(1000)/(sqrt(2)),10`
D. `(50)/(sqrt(2)),0`

Answer

To solve the given problem, we will follow these steps:

### Step 1: Identify the given parameters
The instantaneous EMF and current are given as:
- \( e(t) = 100 \sin(30t) \)
- \( i(t) = 20 \sin(30t - \frac{\pi}{4}) \)

From these equations, we can identify:
- The peak voltage \( E_0 = 100 \) V
- The peak current \( I_0 = 20 \) A

### Step 2: Calculate the phase difference \( \phi \)
The phase difference \( \phi \) between the voltage and current can be determined from the current equation:
- \( i(t) = 20 \sin(30t - \frac{\pi}{4}) \)

Thus, the phase angle \( \phi = \frac{\pi}{4} \) radians.

### Step 3: Calculate the average power consumed in the circuit
The formula for average power \( P \) in an AC circuit is given by:
\[ P = \frac{E_0 I_0}{2} \cos(\phi) \]
Substituting the values we have:
- \( E_0 = 100 \)
- \( I_0 = 20 \)
- \( \phi = \frac{\pi}{4} \) (where \( \cos(\frac{\pi}{4}) = \frac{1}{\sqrt{2}} \))

Now substituting these values into the formula:
\[ P = \frac{100 \times 20}{2} \cdot \cos\left(\frac{\pi}{4}\right) = \frac{2000}{2} \cdot \frac{1}{\sqrt{2}} = 1000 \cdot \frac{1}{\sqrt{2}} = \frac{1000}{\sqrt{2}} \text{ W} \]

### Step 4: Calculate the wattless current
The wattless current \( I_w \) can be calculated using the formula:
\[ I_w = I_{\text{rms}} \cdot \sin(\phi) \]
Where \( I_{\text{rms}} = \frac{I_0}{\sqrt{2}} \).

Calculating \( I_{\text{rms}} \):
\[ I_{\text{rms}} = \frac{20}{\sqrt{2}} \text{ A} \]

Now substituting into the wattless current formula:
\[ I_w = \frac{20}{\sqrt{2}} \cdot \sin\left(\frac{\pi}{4}\right) = \frac{20}{\sqrt{2}} \cdot \frac{1}{\sqrt{2}} = \frac{20}{2} = 10 \text{ A} \]

### Final Answers
- Average Power: \( \frac{1000}{\sqrt{2}} \) W
- Wattless Current: \( 10 \) A