Which of the following is correct for sinusoidal electromagnetic induction :

Options

A. maximum flux, zero emf
B. zero flux , maximum emf
C. zero flux , zero emf
D. (1) & (2) both

Answer

To solve the question regarding sinusoidal electromagnetic induction, we will analyze the relationship between magnetic flux and induced electromotive force (EMF) step by step.

### Step-by-Step Solution:

1. **Understanding the Relationship:**
- The magnetic flux (\( \Phi \)) through a coil is given by the equation:
\[ \Phi(t) = nAB \sin(\omega t) \]
where:
- \( n \) = number of turns in the coil,
- \( A \) = area of the coil,
- \( B \) = magnetic field strength,
- \( \omega \) = angular frequency.

2. **Applying Faraday's Law of Electromagnetic Induction:**
- According to Faraday's law, the induced EMF (\( E \)) is given by:
\[ E = -\frac{d\Phi}{dt} \]
- We need to differentiate the expression for magnetic flux with respect to time \( t \).

3. **Differentiating the Flux Expression:**
- Differentiate \( \Phi(t) \):
\[ E = -\frac{d}{dt}(nAB \sin(\omega t)) = -nAB \omega \cos(\omega t) \]

4. **Analyzing Maximum Flux Condition:**
- The maximum flux occurs when \( \sin(\omega t) = 1 \), which happens at \( \omega t = \frac{\pi}{2} \).
- At this point, \( E \) becomes:
\[ E = -nAB \omega \cos\left(\frac{\pi}{2}\right) = -nAB \omega \cdot 0 = 0 \]
- Thus, when the magnetic flux is at its maximum, the induced EMF is zero.

5. **Analyzing Zero Flux Condition:**
- The flux is zero when \( \sin(\omega t) = 0 \), which occurs at \( \omega t = 0 \) or \( \omega t = \pi \) (and other multiples).
- At this point, \( E \) becomes:
\[ E = -nAB \omega \cos(0) = -nAB \omega \cdot 1 = -nAB \omega \]
- Thus, when the magnetic flux is zero, the induced EMF is at its maximum.

6. **Conclusion:**
- From the above analysis, we conclude:
- When the magnetic flux is maximum, the induced EMF is zero (Statement A is true).
- When the magnetic flux is zero, the induced EMF is maximum (Statement B is true).
- Therefore, both statements A and B are correct.

### Final Answer:
The correct option is D: Both statements 1 and 2 are true.