A Pair of coils of turns `n_(1)` and `n_(2)` are kept close together. Current passing through the first is reduced at r, and emf 3 mV is developed across the other coil. If the second coil carries current which is then reduced at the rate 2 r, the emf produced across the first coil will be

Options

A. `(6n_(1))/(n_(2))mV`
B. `(6n_(2))/(n_(1))mV`
C. 6 mV
D. 3/2 mV

Explanation

To solve the problem, we will use the concept of mutual induction between two coils. The mutual inductance \( M \) relates the rate of change of current in one coil to the induced electromotive force (emf) in the other coil.

### Step-by-Step Solution:

1. **Understanding the Given Data:**
- Let the number of turns in the first coil be \( n_1 \) and in the second coil be \( n_2 \).
- The current in the first coil is reduced at a rate \( r \).
- The induced emf across the second coil (due to the change in current in the first coil) is given as \( \text{emf}_2 = 3 \, \text{mV} \).

2. **Using the Formula for Induced EMF:**
- The formula for the induced emf in the second coil due to the change in current in the first coil is given by:
\[ \text{emf}_2 = -M \frac{di_1}{dt} \]
- Here, \( \frac{di_1}{dt} = -r \) (since the current is reducing).

3. **Relating Induced EMF and Rate of Change of Current:**
- From the above equation, we can write:
\[ 3 \, \text{mV} = M \cdot r \]
- This means that the mutual inductance \( M \) can be expressed as:
\[ M = \frac{3 \, \text{mV}}{r} \]

4. **Considering the Second Coil:**
- Now, when the current in the second coil is reduced at the rate of \( 2r \), we need to find the induced emf across the first coil.
- The induced emf in the first coil due to the change in current in the second coil is given by:
\[ \text{emf}_1 = -M \frac{di_2}{dt} \]
- Here, \( \frac{di_2}{dt} = -2r \) (since the current is reducing).

5. **Calculating the Induced EMF in the First Coil:**
- Substituting the value of \( M \) into the equation for \( \text{emf}_1 \):
\[ \text{emf}_1 = M \cdot (2r) = \left(\frac{3 \, \text{mV}}{r}\right) \cdot (2r) \]
- Simplifying this gives:
\[ \text{emf}_1 = 6 \, \text{mV} \]

6. **Conclusion:**
- Therefore, the induced emf produced across the first coil when the current in the second coil is reduced at the rate of \( 2r \) is \( 6 \, \text{mV} \).

### Final Answer:
The emf produced across the first coil will be **6 mV**.