A circular loop is kept perpendicular to a uniform magnetic field. Loop is rotated about its diameter with constant angular velocity.

Options

A. Magnitude of emf induced will be minimum when magnitude of flux is maximum
B. Magnitude of emf is maximum when magnitude of flux is minimum
C. Phase difference between emf and flux is `(pi)/(2)`
D. None of these

Explanation

To solve the problem of a circular loop rotating in a uniform magnetic field, we will analyze the situation step by step.

### Step 1: Understand the Setup
A circular loop is placed perpendicular to a uniform magnetic field \( B \). The loop is rotated about its diameter with a constant angular velocity \( \omega \).

### Step 2: Define Magnetic Flux
The magnetic flux \( \Phi \) through the loop is given by the formula:
\[ \Phi = B \cdot A \cdot \cos(\theta) \]
where:
- \( B \) is the magnetic field strength,
- \( A \) is the area of the loop,
- \( \theta \) is the angle between the magnetic field and the normal to the surface of the loop.

### Step 3: Determine the Angle \( \theta \)
As the loop rotates, the angle \( \theta \) changes with time. If the loop rotates with an angular velocity \( \omega \), then the angle at time \( t \) can be expressed as:
\[ \theta = \omega t \]

### Step 4: Write the Expression for Flux
Substituting \( \theta \) into the flux equation, we have:
\[ \Phi(t) = B \cdot A \cdot \cos(\omega t) \]

### Step 5: Apply Faraday's Law of Induction
According to Faraday's law, the induced electromotive force (EMF) \( \mathcal{E} \) in the loop is given by the negative rate of change of magnetic flux:
\[ \mathcal{E} = -\frac{d\Phi}{dt} \]

### Step 6: Differentiate the Flux Expression
Now, we differentiate the flux with respect to time:
\[ \mathcal{E} = -\frac{d}{dt}(B \cdot A \cdot \cos(\omega t)) = BA \cdot \omega \cdot \sin(\omega t) \]

### Step 7: Analyze the Relationship Between EMF and Flux
From the expressions derived:
- The magnetic flux \( \Phi(t) = BA \cos(\omega t) \)
- The induced EMF \( \mathcal{E} = BA \omega \sin(\omega t) \)

### Step 8: Determine Maximum and Minimum Values
1. The maximum value of the magnetic flux occurs when \( \cos(\omega t) = 1 \) (i.e., \( \Phi_{\text{max}} = BA \)).
2. The induced EMF is maximum when \( \sin(\omega t) = 1 \) (i.e., \( \mathcal{E}_{\text{max}} = BA \omega \)).
3. The maximum flux occurs when the EMF is zero, and vice versa.

### Step 9: Phase Difference
The sine and cosine functions are complementary:
- \( \sin(\omega t) \) reaches its maximum when \( \cos(\omega t) \) is zero, indicating a phase difference of \( \frac{\pi}{2} \) radians (or 90 degrees) between the EMF and the magnetic flux.

### Conclusion
Based on the analysis, we can conclude:
1. The magnitude of EMF induced will be minimum when the magnitude of flux is maximum.
2. The magnitude of EMF is maximum when the magnitude of flux is minimum.
3. The phase difference between EMF and flux is \( \frac{\pi}{2} \).

### Final Answer
Thus, the correct options are:
- A: Magnitude of EMF induced will be minimum when magnitude of flux is maximum.
- B: Magnitude of EMF is maximum when magnitude of flux is minimum.
- C: Phase difference between the EMF and flux is \( \frac{\pi}{2} \).