The correct answer is $\frac{{\text{F}}}{{2{\text{A}}}}$.
The shear stress is defined as the force per unit area acting parallel to the surface. The maximum shear stress occurs at the neutral axis, which is the axis of the beam that does not experience any bending. The shear force is the force that acts parallel to the cross-section of the beam and causes it to deform. The area of the cross-section is the area of the rectangle.
The shear stress can be calculated using the following equation:
$$\tau = \frac{F}{A}$$
where:
- $\tau$ is the shear stress
- $F$ is the shear force
- $A$ is the area of the cross-section
Substituting in the values for $F$ and $A$, we get:
$$\tau = \frac{{\text{F}}}{{2{\text{A}}}}$$
Therefore, the maximum magnitude of shear stress due to shear force $F$ on a rectangular section of area $A$ at the neutral axis is $\frac{{\text{F}}}{{2{\text{A}}}}$.
Option A is incorrect because it does not take into account the area of the cross-section. Option B is incorrect because it does not take into account the shear force. Option C is incorrect because it does not take into account the neutral axis. Option D is incorrect because it does not take into account the shear stress.