Animated Solution for Physics - Optics: A quarter cylinder of radius $R$ and refractive index $1.5$ is placed on a table. A point object $P$ is kept at a distance of $mR$ from it. Find the value of $m$ for which a ray from $P$ will emerge parallel to the table as shown in figure.

Imagine a ray of light originating from point $P$. It embarks on a fascinating journey, first striking the flat vertical face of the quarter cylinder, and then encountering the curved face. The problem states a crucial condition: the final ray emerges parallel to the table. In the language of optics, a ray traveling parallel to the principal axis means that the final image is formed at infinity ($v=\infty$). To unravel the mystery of the distance $mR$, we need to apply the paraxial refraction formula for both surfaces, step by step.

First, let's analyze the refraction at the flat vertical surface. The general formula for refraction at a spherical surface is:

For a perfectly flat plane surface, the radius of curvature $R$ is infinity. The light travels from air (where ${\mu }_1=1$) into the glass cylinder (where ${\mu }_2=1.5$). The object $P$ is located at a distance of $mR$ to the left of the surface, so following our sign convention, $u=-mR$.

Plugging these values into our formula:

Since any finite number divided by infinity is zero, the right side vanishes:

This tells us that the first image, $I_1$, is a virtual image formed at a distance of $1.5mR$ to the left of the plane surface. This virtual image will now act as the object for the second refraction event.

Now, let's shift our focus to the curved surface. The virtual image $I_1$ is our new object. The pole of this curved surface is at a distance $R$ from the flat face. Therefore, the total object distance $u_2$ from the curved surface's pole is the sum of these distances, making it $u_2=-(1.5mR+R)$.

Furthermore, the center of curvature for this curved face lies at the origin (the flat face), which is to the left of the pole. Hence, the radius of curvature is $R_2=-R$.

This time, light is traveling from the glass (${\mu }_1=1.5$) back into the air (${\mu }_2=1$). We also know the final ray is parallel to the axis, so the final image distance $v_2=\infty$. Let's substitute all these carefully determined values into the refraction formula:

The $\frac{1}{\infty }$ term becomes zero, and the negative signs cancel out beautifully:

We can cancel $R$ from the denominators on both sides, leaving us with a simple linear equation:

Multiplying both sides by 2 to clear the decimal gives:

Subtracting 1 from 3 yields:

Finally, dividing by 1.5, we arrive at our answer:

By methodically tracking the image from one surface to the next, we've successfully determined the exact position the object must be placed to achieve the parallel emergent ray.