The phase difference between the flux linkage and the induced e.m.f. in a rotating coil in a uniform magnetic field

Options

A. `pi//2`
B. `pi//3`
C. `-pi//6`
D. `pi`

Explanation

To solve the problem of finding the phase difference between the flux linkage and the induced e.m.f. in a rotating coil in a uniform magnetic field, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Flux Linkage**:
The magnetic flux (Φ) linked with a coil rotating in a uniform magnetic field can be expressed as:
\[ \Phi = B \cdot A \cdot \cos(\theta) \]
where \(B\) is the magnetic field strength, \(A\) is the area of the coil, and \(\theta\) is the angle between the magnetic field and the normal to the coil.

2. **Express the Angle in Terms of Time**:
If the coil is rotating with an angular velocity \(\omega\), the angle \(\theta\) can be expressed as:
\[ \theta = \omega t \]
Thus, the flux can be rewritten as:
\[ \Phi = B \cdot A \cdot \cos(\omega t) \]

3. **Calculate the Induced e.m.f.**:
The induced e.m.f. (ε) in the coil can be found using Faraday's law of electromagnetic induction, which states:
\[ \varepsilon = -\frac{d\Phi}{dt} \]
Substituting the expression for flux:
\[ \varepsilon = -\frac{d}{dt}(B \cdot A \cdot \cos(\omega t)) \]

4. **Differentiate the Flux**:
Using the chain rule to differentiate:
\[ \varepsilon = -B \cdot A \cdot \frac{d}{dt}(\cos(\omega t)) = -B \cdot A \cdot (-\omega \sin(\omega t)) \]
Simplifying gives:
\[ \varepsilon = B \cdot A \cdot \omega \sin(\omega t) \]

5. **Identify the Phase of Flux and e.m.f.**:
The expressions for flux and induced e.m.f. are:
- Flux: \(\Phi = B \cdot A \cdot \cos(\omega t)\)
- Induced e.m.f.: \(\varepsilon = B \cdot A \cdot \omega \sin(\omega t)\)

The cosine function can be expressed in terms of sine:
\[ \cos(\omega t) = \sin\left(\omega t + \frac{\pi}{2}\right) \]
This shows that the flux is leading the induced e.m.f. by \(\frac{\pi}{2}\).

6. **Calculate the Phase Difference**:
The phase difference (\(\Delta \phi\)) between the flux linkage and the induced e.m.f. is given by:
\[ \Delta \phi = \left(\omega t + \frac{\pi}{2}\right) - \omega t = \frac{\pi}{2} \]

### Final Answer:
The phase difference between the flux linkage and the induced e.m.f. in a rotating coil in a uniform magnetic field is \(\frac{\pi}{2}\).

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