D

Step 1: Consider a uniformly charged infinite line of charge with a linear charge density $\lambda = q$. The electric field (E) generated by an infinite line charge is given by Gauss's Law.
Step 2: To find the electric field at a distance $r$ from the axis of the line charge, we consider a cylindrical Gaussian surface of radius $r$ and length $L$.
Using Gauss's law, we have:
$$\Phi_E = E \cdot A = E \cdot (2\pi r L)$$
where $\Phi_E$ is the electric flux, $E$ is the electric field, and $A$ is the surface area of the cylindrical surface.
Step 3: The charge enclosed ($Q_{enc}$) by the Gaussian surface is:
$$Q_{enc} = \lambda L = q L$$
Step 4: According to Gauss's Law:
$$\Phi_E = \frac{Q_{enc}}{\epsilon_0}$$
Therefore,
$$E \cdot (2\pi r L) = \frac{q L}{\epsilon_0}$$
Step 5: Rearranging for E gives:
$$E = \frac{q}{2\pi \epsilon_0 r}$$
Step 6: This shows that the electric field E is inversely proportional to the distance r from the wire.
Therefore, the correct option is: Option D: inversely proportional to r².