A fan blade of length 2a rotates with frequency f cycles per second perpendicular to a magnetic field B. Find the p.d between the centre and the end of the blade.

Answer

To find the potential difference (p.d.) between the center and the end of a rotating fan blade in a magnetic field, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Given Information:**
- Length of the fan blade = 2a
- Frequency of rotation = f (cycles per second)
- Magnetic field = B (perpendicular to the plane of rotation)

2. **Identify the Area Swept by the Blade:**
- The fan blade rotates around its center, so the distance from the center to the end of the blade is 'a'.
- The area swept by the blade during one complete rotation (one cycle) is given by the formula for the area of a circle:
\[ \text{Area} = \pi r^2 = \pi a^2 \]

3. **Calculate the Time Period (T):**
- The time period (T) is the reciprocal of frequency (f):
\[ T = \frac{1}{f} \]

4. **Use Faraday's Law of Electromagnetic Induction:**
- According to Faraday's law, the electromotive force (emf, ε) induced in a circuit is equal to the rate of change of magnetic flux through the circuit:
\[ \epsilon = -\frac{d\Phi}{dt} \]
- The magnetic flux (Φ) through the area swept by the blade is:
\[ \Phi = B \cdot \text{Area} = B \cdot \pi a^2 \]

5. **Calculate the Rate of Change of Magnetic Flux:**
- The rate of change of magnetic flux as the blade rotates is:
\[ \frac{d\Phi}{dt} = B \cdot \frac{d(\pi a^2)}{dt} \]
- Since the area is constant, we can express the change in time as:
\[ \frac{d\Phi}{dt} = B \cdot \pi a^2 \cdot \frac{1}{T} = B \cdot \pi a^2 \cdot f \]

6. **Calculate the Induced EMF (p.d.):**
- Substituting this into Faraday's law gives:
\[ \epsilon = -B \cdot \pi a^2 \cdot f \]
- The negative sign indicates the direction of the induced emf according to Lenz's law, but for the magnitude of the potential difference (p.d.), we can ignore the negative sign:
\[ \text{p.d.} = B \cdot \pi a^2 \cdot f \]

### Final Answer:
The potential difference (p.d.) between the center and the end of the blade is:
\[ \text{p.d.} = B \cdot \pi a^2 \cdot f \]