Define Coefficient of Coupling

The coefficient of coupling $k$ represents the fraction of the magnetic flux generated by one coil that links with the other coil.

$k = \frac{\phi_{12}}{\phi_1}$

Analyze the condition

The problem states that all the flux produced by coil 1 ($\phi_1$) links with coil 2 ($\phi_{12}$). Therefore, the flux linkage $\phi_{12}$ is equal to $\phi_1$.

$\phi_{12} = \phi_1$

Calculate the result

Substituting the condition into the definition of $k$, we get the value of the coefficient of coupling.

$k = \frac{\phi_1}{\phi_1} = 1$