A conducting loop rotates with constant angular velocity about its fixed diameter in a uniform magnetic field, whose direction is perpendicular to that fixed diameter.

Options

A. (a) The emf will be maximum at the moment when flux is zero
B. (b) The emf will be `0` at the moment when flux ismaximum
C. ( c) The emf will be maximum at the moment when plane of the loop is parallel to the magnetic field
D. (d) The phase difference between the flux and the emf `pi//2`

Explanation

To solve the problem of a conducting loop rotating in a uniform magnetic field, we will follow these steps:

### Step 1: Understand the Setup
We have a conducting loop that rotates with a constant angular velocity (\(\omega\)) about its fixed diameter. The magnetic field (\(B\)) is uniform and perpendicular to the fixed diameter of the loop.

### Step 2: Define the Area Vector
Initially, the area vector (\(A\)) of the loop is aligned with the magnetic field. As the loop rotates by an angle \(\theta\), the area vector will also rotate, and the angle between the area vector and the magnetic field will change.

### Step 3: Express the Angle
The angle \(\theta\) can be expressed as:
\[ \theta = \omega t \]
where \(t\) is the time.

### Step 4: Calculate Magnetic Flux
The magnetic flux (\(\Phi\)) through the loop at any time \(t\) is given by:
\[ \Phi = B \cdot A \cdot \cos(\theta) \]
Substituting \(\theta\):
\[ \Phi(t) = B \cdot A \cdot \cos(\omega t) \]

### Step 5: Determine Induced EMF
The induced electromotive force (EMF) (\(\mathcal{E}\)) in the loop can be calculated using Faraday's law of electromagnetic induction:
\[ \mathcal{E} = -\frac{d\Phi}{dt} \]
Calculating the derivative:
\[ \mathcal{E} = -\frac{d}{dt}(B \cdot A \cdot \cos(\omega t)) = B \cdot A \cdot \omega \cdot \sin(\omega t) \]

### Step 6: Analyze Maximum and Minimum EMF
1. The EMF will be maximum when \(\sin(\omega t) = 1\), which occurs when \(\omega t = \frac{\pi}{2} + n\pi\) (where \(n\) is an integer).
2. The EMF will be zero when \(\sin(\omega t) = 0\), which occurs when \(\omega t = n\pi\).

### Step 7: Analyze Flux
1. The magnetic flux will be maximum when \(\cos(\omega t) = 1\), which occurs when \(\omega t = 0 + 2n\pi\).
2. The magnetic flux will be zero when \(\cos(\omega t) = 0\), which occurs when \(\omega t = \frac{\pi}{2} + n\pi\).

### Step 8: Determine Phase Difference
The flux (\(\Phi\)) is given by \(B \cdot A \cdot \cos(\omega t)\) and the induced EMF is given by \(B \cdot A \cdot \omega \cdot \sin(\omega t)\). The sine function leads the cosine function by \(\frac{\pi}{2}\) radians (or 90 degrees). Thus, the phase difference between the flux and the EMF is \(\frac{\pi}{2}\).

### Conclusion
Based on the analysis:
1. The EMF is maximum when the magnetic flux is zero.
2. The EMF is zero when the magnetic flux is maximum.
3. The EMF is maximum when the plane of the loop is parallel to the magnetic field.
4. The phase difference between the flux and the EMF is \(\frac{\pi}{2}\).