If the current in the primary circuit of a pair of coils changes from 10 A to 0 in 0.1 s, calculate (i) the induced emf in the secondary if the mutual inductance between the two coils is 2 H and (ii) the change of flux per turn in the secondary if it has 500 turns.

Answer

To solve the given problem step by step, we will address both parts of the question separately.

### Part (i): Calculate the induced emf in the secondary coil.

1. **Identify the given values:**
- Initial current in the primary circuit, \( I_1 = 10 \, \text{A} \)
- Final current in the primary circuit, \( I_2 = 0 \, \text{A} \)
- Time interval, \( \Delta t = 0.1 \, \text{s} \)
- Mutual inductance, \( M = 2 \, \text{H} \)

2. **Calculate the change in current (\( \Delta I \)):**
\[ \Delta I = I_2 - I_1 = 0 - 10 = -10 \, \text{A} \]

3. **Calculate the rate of change of current (\( \frac{dI}{dt} \)):**
\[ \frac{dI}{dt} = \frac{\Delta I}{\Delta t} = \frac{-10 \, \text{A}}{0.1 \, \text{s}} = -100 \, \text{A/s} \]

4. **Use the formula for induced emf (\( \epsilon \)):**
\[ \epsilon = -M \frac{dI}{dt} \]
Substituting the values:
\[ \epsilon = -2 \, \text{H} \times (-100 \, \text{A/s}) = 200 \, \text{V} \]

### Part (ii): Calculate the change of flux per turn in the secondary coil.

1. **Identify the number of turns in the secondary coil:**
- Number of turns, \( N = 500 \)

2. **Use the relationship between induced emf and change in magnetic flux:**
\[ \epsilon = -N \frac{d\Phi}{dt} \]

3. **Rearranging the equation gives us:**
\[ N \frac{d\Phi}{dt} = M \frac{dI}{dt} \]

4. **Substituting the values:**
\[ N \frac{d\Phi}{dt} = 2 \, \text{H} \times (-100 \, \text{A/s}) = -200 \, \text{Wb/s} \]

5. **Now, calculate \( \frac{d\Phi}{dt} \):**
\[ \frac{d\Phi}{dt} = \frac{-200 \, \text{Wb/s}}{500} = -0.4 \, \text{Wb/turn} \]

### Final Answers:
- (i) The induced emf in the secondary is \( 200 \, \text{V} \).
- (ii) The change of flux per turn in the secondary is \( -0.4 \, \text{Wb/turn} \).