Magnetic Fields from Wire Halves

An infinitely long circular wire with radius $R$ carries a current into the page which is uniformly distributed over the cross sectional area. Consider the total magnetic flux density at a test point located a distance $2 R$ from the center of the wire. The total field can be decomposed into contributions from the closer half and farther half of the wire.

$\large{\vec{B_T} = \vec{B_C} + \vec{B_F}}$

What is the ratio of the magnitude of $\vec{B_C}$ to the magnitude of $\vec{B_F}$ ?

$\large{\frac{|\vec{B_C}|}{|\vec{B_F}|} = ?}$