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

Options

A. a.induced emf in the inductor
B. b.Current
C. c.Both a and b
D. d.Neither a nor b

Answer

To determine which quantities have a zero average value over a cycle in an AC circuit with an inductor-coil that has some resistance, we can analyze the behavior of the induced emf and the current in the circuit.

### Step-by-Step Solution:

1. **Understanding the AC Circuit**:
- In an AC circuit, the voltage (V) can be expressed as a sinusoidal function. For example, we can write:
\[ V(t) = V_m \sin(\omega t) \]
- Here, \(V_m\) is the maximum voltage, \(\omega\) is the angular frequency, and \(t\) is time.

2. **Current in the Inductor**:
- The current (I) in an inductor connected to an AC source lags behind the voltage. The current can be expressed as:
\[ I(t) = I_m \cos(\omega t) \]
- Alternatively, it can also be represented as:
\[ I(t) = I_n \sin(\omega t) \]
- The choice of sine or cosine depends on the phase relationship, but both forms will yield similar results regarding average values.

3. **Calculating the Average Value**:
- The average value of a sinusoidal function over one complete cycle (from \(0\) to \(2\pi\)) is calculated using the integral:
\[ \text{Average} = \frac{1}{T} \int_0^T f(t) dt \]
- For \(f(t) = \sin(\omega t)\) or \(f(t) = \cos(\omega t)\), the average value over one complete cycle is:
\[ \text{Average} = \frac{1}{2\pi} \int_0^{2\pi} \sin(\omega t) dt = 0 \]
- Similarly, for the cosine function:
\[ \text{Average} = \frac{1}{2\pi} \int_0^{2\pi} \cos(\omega t) dt = 0 \]

4. **Conclusion**:
- Since both the induced emf (which varies sinusoidally) and the current (which also varies sinusoidally) have an average value of zero over a complete cycle, we conclude that:
- Both the induced emf and the current have a zero average value over a cycle.

5. **Final Answer**:
- Therefore, the correct option is:
- **C) Both A and B** (induced emf and current both have zero average value over a cycle).