Question

Question 22:

Consider refraction of parallel rays (also non paraxial rays) to be converging at single point & for reflection as well

• A. If parallel rays are from rarer to denser medium then it should be ellipsoid surface separating mediums.
• B. If parallel ray are from denser to rarer medium then it should hyperboloid surface separating medium
• C. If parallel rays are reflected to a common point then mirror should be parabolic mirror
• D. If parallel rays are reflected to a common point then mirror should be hyperbolic mirror

Answer 1

Answer: (A), (B), (C)

A, B, C

Answer 2

The problem asks us to identify the correct statements regarding the shapes of optical surfaces (refracting or reflecting) that cause parallel rays to converge to a single point, considering non-paraxial rays as well. This implies that we need to consider the exact geometrical shapes derived from Fermat's principle (constant optical path length).

A. If parallel rays are from rarer to denser medium then it should be ellipsoid surface separating mediums.

Consider parallel rays incident from a rarer medium (refractive index $n_1$) onto a surface separating it from a denser medium (refractive index $n_2 > n_1$). If these rays converge to a single point (focus) F in the denser medium, then according to Fermat's principle, the optical path length (OPL) from any point on an incident wavefront to the focus F must be constant.

Let the incident parallel rays be along the x-axis. Let the surface be at point P(x, y) and the focus be F(f, 0). The OPL from a reference plane $x=x_0$ (perpendicular to the incident rays) to F is given by: $OPL = n_1(x_0 - x) + n_2 \sqrt{(x-f)^2 + y^2} = \text{constant}$

Rearranging and squaring, we get an equation of the form: $(n_2^2 - n_1^2)x^2 - (2n_2^2 f + 2C'n_1)x + n_2^2 y^2 + (n_2^2 f^2 - C'^2) = 0$ where $C'$ is a constant related to the total OPL.

The coefficient of $x^2$ is $(n_2^2 - n_1^2)$ and the coefficient of $y^2$ is $n_2^2$. Since $n_2 > n_1$, both coefficients are positive. This indicates that the surface is an ellipsoid (or a sphere, a special case of ellipsoid). This is a well-known property of aplanatic surfaces for refraction.

Thus, statement A is correct.

B. If parallel ray are from denser to rarer medium then it should hyperboloid surface separating medium.

Similar to case A, if parallel rays are from a denser medium ($n_1$) to a rarer medium ($n_2 < n_1$), and they converge to a single point F, the OPL equation remains the same: $OPL = n_1(x_0 - x) + n_2 \sqrt{(x-f)^2 + y^2} = \text{constant}$

The resulting equation for the surface is again: $(n_2^2 - n_1^2)x^2 - (2n_2^2 f + 2C'n_1)x + n_2^2 y^2 + (n_2^2 f^2 - C'^2) = 0$

In this case, since $n_1 > n_2$, the coefficient of $x^2$, $(n_2^2 - n_1^2)$, is negative, while the coefficient of $y^2$, $n_2^2$, is positive. This indicates that the surface is a hyperboloid.

Thus, statement B is correct.

C. If parallel rays are reflected to a common point then mirror should be parabolic mirror.

For reflection, according to Fermat's principle, the optical path length from any point on an incident wavefront to the focus F must be constant.

Let parallel rays be incident along the x-axis onto a mirror surface P(x, y), and let them converge to a focus F(f, 0). The OPL from a reference plane $x=x_0$ to F is: $OPL = (x_0 - x) + \sqrt{(x-f)^2 + y^2} = \text{constant}$

Rearranging and squaring: $\sqrt{(x-f)^2 + y^2} = C' + x$ (where $C'$ is a constant) $(x-f)^2 + y^2 = (C'+x)^2$ $x^2 - 2xf + f^2 + y^2 = C'^2 + 2C'x + x^2$ $-2xf + f^2 + y^2 = C'^2 + 2C'x$ $y^2 = (2f + 2C')x + (C'^2 - f^2)$

This is the equation of a parabola ($y^2 = Ax + B$). This is a fundamental property of parabolic mirrors: parallel rays incident on a parabolic mirror converge to its focus.

Thus, statement C is correct.

D. If parallel rays are reflected to a common point then mirror should be hyperbolic mirror.

A hyperbolic mirror does not reflect parallel rays to a single point. Its property is that rays directed towards one focus are reflected such that they appear to come from the other focus (for convex side reflection) or converge to the other focus (for concave side reflection). Parabolic mirrors are used to focus parallel rays to a single point.

Thus, statement D is incorrect.

Therefore, statements A, B, and C are correct.

The final answer is $\boxed{A, B, C}$

Explanation of the solution: The solution relies on the principle of constant optical path length (Fermat's principle).

1. Refraction (Rarer to Denser): For parallel rays from a rarer medium ($n_1$) to converge to a point in a denser medium ($n_2 > n_1$), the optical path length ($n_1 \times \text{path in rarer} + n_2 \times \text{path in denser}$) must be constant. This condition mathematically defines an ellipsoid.
2. Refraction (Denser to Rarer): Similarly, for parallel rays from a denser medium ($n_1$) to converge to a point in a rarer medium ($n_2 < n_1$), the constant optical path length condition defines a hyperboloid.
3. Reflection: For parallel rays to be reflected to a common point (focus), the path length from the incident wavefront to the focus must be constant. This condition defines a parabola.