Two coils 'P' and 'Q' are separated by some distance. When a...

Since the coils are co-axial, the flux through the coils can be written as the product of their surface area and the magnetic field produced by the coil along its axis.

Consider the flux passing through Q

${ϕ}_{Q} = \dfrac{{\text{\mu}}_{0}{i}_{1}{R}^{2}}{2\left({R}^{2} + {x}^{2}\right)\dfrac{3}{2}} \times \text{\pi}{R}^{2} = {10}^{- 3}Wb$

And the flux passing through P:
${ϕ}_{P} = \dfrac{{\text{\mu}}_{0}{i}_{1}{r}^{2}}{2\left({R}^{2} + {x}^{2}\right)\dfrac{3}{2}} \times \text{\pi}{R}^{2}$

Taking the ratio
$\dfrac{{ϕ}_{P}}{{ϕ}_{Q}} = \dfrac{{i}_{2}}{{i}_{1}} \times \dfrac{{\left({R}^{2} + {x}^{2}\right)}^{\dfrac{3}{2}}}{{\left({R}^{2} + {x}^{2}\right)}^{\dfrac{3}{2}}} ⇒ \dfrac{{ϕ}_{P}}{{10}^{- 3}} = \dfrac{{i}_{2}}{{i}_{1}}$
$⇒ \dfrac{{ϕ}_{P}}{{10}^{- 3}} = \dfrac{2}{3}$
$∴ {ϕ}_{P} = 6.67 \times {10}^{- 4}Wb$