In any `AC` circuit the emf `(e)` and the current `(i)` at any instant are given respectively by `e= V_(0)sin omega t`
`i=I_(0) sin (omegat-phi)`
The average power in the circuit over one cycle of `AC` is

Options

A. `(V_(0)i_(0))/(2)`
B. `(V_(0)i_(0))/(2)sin phi`
C. `(V_(0)i_(0))/(2)cos phi`
D. `V_(0)i_(0)`

Answer

To find the average power in an AC circuit over one cycle, we can follow these steps:

### Step 1: Understand the given equations
We are given the equations for the electromotive force (emf) and the current in the circuit:
- \( e = V_0 \sin(\omega t) \)
- \( i = I_0 \sin(\omega t - \phi) \)

Where:
- \( V_0 \) = peak voltage (emf)
- \( I_0 \) = peak current
- \( \phi \) = phase difference between the emf and the current

### Step 2: Calculate the RMS values
The root mean square (RMS) values for the emf and current are calculated as follows:
- The RMS value of the voltage \( e \) is given by:
\[ e_{\text{rms}} = \frac{V_0}{\sqrt{2}} \]
- The RMS value of the current \( i \) is given by:
\[ i_{\text{rms}} = \frac{I_0}{\sqrt{2}} \]

### Step 3: Use the formula for average power
The average power \( P_{\text{avg}} \) in an AC circuit can be calculated using the formula:
\[ P_{\text{avg}} = e_{\text{rms}} \cdot i_{\text{rms}} \cdot \cos(\phi) \]

### Step 4: Substitute the RMS values into the power formula
Substituting the RMS values we calculated into the power formula:
\[ P_{\text{avg}} = \left(\frac{V_0}{\sqrt{2}}\right) \cdot \left(\frac{I_0}{\sqrt{2}}\right) \cdot \cos(\phi) \]

### Step 5: Simplify the expression
Now simplifying the expression:
\[ P_{\text{avg}} = \frac{V_0 I_0}{2} \cdot \cos(\phi) \]

### Final Result
Thus, the average power in the circuit over one cycle of AC is:
\[ P_{\text{avg}} = \frac{V_0 I_0}{2} \cos(\phi) \]