Which of the following quantities have zero average value over a cycle.If an inductor coil having some resistance is connected to a sinusoidal `AC` source.

Options

A. induced `emf` in the inductor
B. current
C. joule heat
D. magnetic energy stored in the inductor

Explanation

To determine which quantities have a zero average value over a cycle when an inductor coil with some resistance is connected to a sinusoidal AC source, we can analyze the behavior of current and voltage in an AC circuit.

### Step-by-Step Solution:

1. **Understanding AC Voltage and Current**:
- The voltage across the AC source can be expressed as \( V(t) = V_0 \sin(\omega t) \), where \( V_0 \) is the peak voltage and \( \omega \) is the angular frequency.
- The current \( I(t) \) through the circuit can be derived from the voltage and the impedance \( Z \) of the circuit.

2. **Impedance of the Circuit**:
- The impedance \( Z \) of an inductor with resistance is given by:
\[ Z = R + jL\omega \]
- Here, \( R \) is the resistance, \( L \) is the inductance, and \( j \) is the imaginary unit.

3. **Current Calculation**:
- The current \( I(t) \) can be expressed as:
\[ I(t) = \frac{V(t)}{Z} \]
- Since \( Z \) is a complex quantity, the current will also be a sinusoidal function, but it may be phase-shifted.

4. **Average Value of Sine Function**:
- The average value of a sine function over one complete cycle is zero:
\[ \text{Average}(V) = \frac{1}{T} \int_0^T V(t) dt = 0 \]
- This applies to both the voltage \( V(t) \) and the current \( I(t) \).

5. **Average Values of Current and Voltage**:
- Since \( I(t) \) is proportional to \( V(t) \), we can conclude:
\[ I_{\text{avg}} = k \cdot V_{\text{avg}} = k \cdot 0 = 0 \]
- Similarly, the voltage across the inductor \( V_L(t) \) will also have an average value of zero.

6. **Conclusion**:
- Therefore, the average values of the following quantities over a complete cycle are zero:
- Average current \( I_{\text{avg}} = 0 \)
- Average voltage across the inductor \( V_L = 0 \)

### Final Answer:
The quantities that have zero average value over a cycle are:
- Average current \( I_{\text{avg}} \)
- Average voltage across the inductor \( V_L \)