Two rays are incident on a spherical mirror of radius \[{\rm{R = 5cm}}\] parallel to its optical axis at distances \[{{\rm{h}}_1}{\rm{ = 0}}{\rm{.5cm}}\] and \[{{\rm{h}}_2}{\rm{ = 3cm}}\]. Determine the distance \[\Delta x\] between the points at which these rays intersect the optical axis after being reflected at the mirror.

Hint: The paraxial rays go through the region \[\dfrac{{\rm{R}}}{2}\] from the concave mirror's pole according to the query. Given that the marginal rays experience greater deviation than the paraxial rays, the solution to the aforementioned issue can be generated with ease. We shall therefore observe how the marginal and paraxial rays' positions with respect to the concave mirror's pole change as a result of the rays' deviation.



Complete answer:

Let \[{\rm{ABC}}\] be the ray incident at a distance \[{\rm{BE}}\] from the mirror axis, and let \[{\rm{O}}\] be the centre of the mirror's spherical surface.

The triangle \[{\rm{OBC}}\] is isosceles because according to the law of reflection \[\angle {\rm{ABO}} = \angle {\rm{OBC}}\] and \[\angle {\rm{BOC}} = \angle {\rm{ABO}}\] are alternate-interior angles, which can be seen from the right triangle \[{\rm{OBE}}\]. Hence, we have the triangle \[{\rm{ODC}}\] with \[{\rm{OD = DB = }}\dfrac{{\rm{R}}}{2}\].
Now from triangle ODC,
\[{\rm{x = }}\dfrac{R}{{2\cos \alpha }}{\rm{ = }}\dfrac{R^2}{{2\sqrt {{R^2} - {h^2}} }}\]
We know that the ray reflected by the mirror and the optical axis is said to be the point of intersection \[{\rm{C}}\].
The distance \[{{\rm{x}}_1} \approx \dfrac{{\rm{R}}}{2}\] with an error of about \[0.5\% \] since \[{{\rm{h}}_1}^2{\rm{ < < }}{{\rm{R}}^2}\] is for a ray propagating at a distance \[{{\rm{h}}_1}\]
Also, it is to be understand that, for a ray propagating at a distance \[{{\rm{h}}_2}\] the distance \[{{\rm{x}}_2} = 3.125\;{\rm{cm}}\]
From the previous calculations, finally, we obtain
\[\Delta {\rm{x}} = {{\rm{x}}_2} - {{\rm{x}}_1}\]
On substituting the value from the given data, we get
\[ \simeq 0.6\;{\rm{cm}}\]
Therefore, the distance \[\Delta x\] between the points at which these rays intersect the optical axis after being reflected at the mirror is \[0.6\;{\rm{cm}}\]


Note:It should be mentioned that this phenomenon is known as spherical aberration. The aperture of the spherical mirrors is the only thing that causes spherical aberration. The spherical mirrors will have the least spherical aberration if the aforementioned criteria are satisfied.