A circular conducting ring is rotated about one its diameter in a magnetic field

Options

A. If magnetic field is uniform, no emf will be induced
B. If magnetic field is uniform and its induction increases with time, only them emf is induced
C. If magnetic field is uniform and induction of magnetic field is increasing at a constant rate, emf induced in ring may zero for an elemental time interval
D. None of these

Answer

To solve the problem of a circular conducting ring being rotated about one of its diameters in a magnetic field, we need to analyze how the magnetic flux through the ring changes as it rotates. Here’s a step-by-step solution:

### Step 1: Understand the Setup
We have a circular conducting ring that is rotated about one of its diameters in a magnetic field. The magnetic field can be uniform or non-uniform.

**Hint:** Visualize the ring and the magnetic field lines to understand their orientation relative to the ring.

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

**Hint:** Remember that the area of the ring remains constant, but the angle \(θ\) changes as the ring rotates.

### Step 3: Analyze the Rotation
As the ring rotates about its diameter, the angle \(θ\) changes continuously. When the plane of the ring is perpendicular to the magnetic field, \(θ = 0°\) and the flux is maximum. When the plane of the ring is parallel to the magnetic field, \(θ = 90°\) and the flux is zero.

**Hint:** Think about how the orientation of the ring affects the angle \(θ\) and thus the magnetic flux.

### Step 4: Induced EMF Calculation
According to Faraday's law of electromagnetic induction, the induced EMF (ε) in the ring is given by the rate of change of magnetic flux:
\[ ε = -\frac{dΦ}{dt} \]
As the ring rotates, \(Φ\) changes with time, leading to a non-zero induced EMF.

**Hint:** Consider how the change in angle affects the rate of change of flux over time.

### Step 5: Conclusion
If the magnetic field is uniform and the ring is rotating, there will be an induced EMF due to the changing magnetic flux. If the magnetic field is non-uniform, the situation becomes more complex, but the principle remains the same: a change in flux leads to induced EMF.

**Hint:** Reflect on the conditions under which EMF is induced and how they relate to the uniformity of the magnetic field.

### Final Answer
In conclusion, as the circular conducting ring rotates in a magnetic field, an EMF will be induced due to the changing magnetic flux through the ring.

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