D

Step 1: The electric field due to an electric dipole along its axis at a distance r is given by the formula:
$$ E = \frac{1}{4\pi \epsilon_0} \cdot \frac{2p}{r^3} $$
where p is the dipole moment.

Step 2: When the dipole is rotated through 90°, it lies in the perpendicular direction to the field point. The electric field due to the dipole at this new orientation can be described by the formula for the electric field perpendicular to the dipole axis:
$$ E' = \frac{1}{4\pi \epsilon_0} \cdot \frac{p}{r^3} $$

This implies that the electric field intensity in this orientation is half of that on the axis, i.e.,
$$ E' = \frac{E}{2} $$

Therefore, the electric field intensity at the same point after rotating the dipole through 90° will be:
$$ E' = \frac{E}{2}. $$
Thus, the correct answer is D: \( \frac{E}{2} \).