A magnet is brought towards a coil (i) speedily (ii) slowly then the induced e.m.f.//induced charge will be respectively

Options

A. more in first case/more in first case
B. more in first case/equal in both case
C. less in first case/more in second case
D. less in first case/equal in both case

Explanation

To solve the problem, we need to analyze the situation where a magnet is brought towards a coil at two different speeds: speedily and slowly. We will determine the induced electromotive force (e.m.f.) and the induced charge in each case.

### Step-by-Step Solution:

1. **Understanding Faraday's Law of Electromagnetic Induction**:
- According to Faraday's law, the induced e.m.f. (ε) in a coil is given by the formula:
\[ \varepsilon = -\frac{d\Phi}{dt} \]
where \( \Phi \) is the magnetic flux through the coil.

2. **Magnetic Flux Change**:
- Magnetic flux (\( \Phi \)) is defined 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's surface.

3. **Case (i): Magnet Brought Speedily**:
- When the magnet is brought towards the coil speedily, the rate of change of magnetic flux (\( \frac{d\Phi}{dt} \)) is large. This results in a larger induced e.m.f.:
\[ \varepsilon_{\text{speedy}} > \varepsilon_{\text{slow}} \]
- Therefore, the induced e.m.f. is **greater** when the magnet is brought speedily.

4. **Case (ii): Magnet Brought Slowly**:
- When the magnet is brought towards the coil slowly, the rate of change of magnetic flux is smaller, leading to a smaller induced e.m.f.:
\[ \varepsilon_{\text{slow}} < \varepsilon_{\text{speedy}} \]

5. **Induced Charge Calculation**:
- The induced charge (Q) can be calculated using the formula:
\[ Q = \frac{\varepsilon \cdot t}{R} \]
where \( R \) is the resistance of the coil, and \( t \) is the time for which the e.m.f. acts.
- Since the charge depends on the e.m.f. and the time, we need to consider the time taken in both cases.

6. **Induced Charge for Both Cases**:
- In the case of a speedy approach, the time taken is shorter, but the induced e.m.f. is higher. In the case of a slow approach, the time taken is longer, but the induced e.m.f. is lower.
- However, the total induced charge will be the same in both cases because the product of e.m.f. and time will yield the same result:
\[ Q_{\text{speedy}} = Q_{\text{slow}} \]

### Conclusion:
- Induced e.m.f. when the magnet is brought speedily: **Greater**
- Induced e.m.f. when the magnet is brought slowly: **Lesser**
- Induced charge in both cases: **Equal**

### Final Answer:
- Induced e.m.f.: Greater (speedily), Lesser (slowly)
- Induced charge: Equal in both cases