To solve the problem, we need to find the magnetic flux through coil 'P' when a current of 2A flows through coil 'Q', given that when a current of 3A flows through coil 'P', a magnetic flux of \(10^{-3} \, \text{Wb}\) passes through coil 'Q'.
### Step-by-Step Solution:
1. **Understanding the Relationship**:
The magnetic flux through coil 'Q' when a current flows through coil 'P' is given by the formula:
\[
\Phi_Q = M \cdot I_P
\]
where \(M\) is the mutual inductance between the coils, \(I_P\) is the current in coil 'P', and \(\Phi_Q\) is the magnetic flux through coil 'Q'.
2. **Calculating Mutual Inductance**:
From the problem, when \(I_P = 3 \, \text{A}\), the flux \(\Phi_Q = 10^{-3} \, \text{Wb}\). We can rearrange the formula to find \(M\):
\[
M = \frac{\Phi_Q}{I_P} = \frac{10^{-3} \, \text{Wb}}{3 \, \text{A}} = \frac{10^{-3}}{3} \, \text{H}
\]
3. **Finding the Flux through Coil 'P'**:
Now, we need to find the flux through coil 'P' when a current of \(2 \, \text{A}\) flows through coil 'Q'. The formula for the flux through coil 'P' is:
\[
\Phi_P = M \cdot I_Q
\]
where \(I_Q = 2 \, \text{A}\).
4. **Substituting the Values**:
Substitute the value of \(M\) and \(I_Q\) into the equation:
\[
\Phi_P = \left(\frac{10^{-3}}{3}\right) \cdot 2 = \frac{2 \times 10^{-3}}{3} \, \text{Wb}
\]
5. **Calculating the Final Value**:
Now, calculate \(\Phi_P\):
\[
\Phi_P = \frac{2 \times 10^{-3}}{3} = 0.6667 \times 10^{-3} \, \text{Wb} \approx 6.67 \times 10^{-4} \, \text{Wb}
\]
### Final Answer:
Thus, the magnetic flux through coil 'P' when a current of 2A flows through coil 'Q' is:
\[
\Phi_P \approx 6.67 \times 10^{-4} \, \text{Wb}
\]
