a portion of straight glass rod of diameter 4cm and refractive index 1.5 is bent into a arc of circle and parallel beam of light is incident on it find smallest value of r which permits light to pass through arc..

To determine the smallest radius \( r \) of the arc that allows light to pass through the bent glass rod, we need to consider the principles of refraction and the geometry of the situation. The key concept here is that light will pass through the arc if it does not undergo total internal reflection. This can be analyzed using Snell's Law and the critical angle for the glass-air interface.

Understanding Refraction and Critical Angle

When light travels from a medium with a higher refractive index (like glass) to one with a lower refractive index (like air), it can be refracted or reflected. The critical angle \( \theta_c \) is the angle of incidence above which total internal reflection occurs. This angle can be calculated using the formula:

Critical Angle Formula:

\( \theta_c = \sin^{-1} \left( \frac{n_2}{n_1} \right) \)

Where \( n_1 \) is the refractive index of the glass (1.5) and \( n_2 \) is the refractive index of air (approximately 1.0).

Calculating the Critical Angle

Substituting the values into the formula:

\( \theta_c = \sin^{-1} \left( \frac{1.0}{1.5} \right) \)

\( \theta_c = \sin^{-1} \left( 0.6667 \right) \)

\( \theta_c \approx 41.81^\circ \)

This means that if the angle of incidence exceeds this critical angle, light will not pass through the arc and will instead be totally internally reflected.

Geometry of the Arc

Next, we need to relate this critical angle to the geometry of the arc. The light beam is incident parallel to the axis of the rod, and as it enters the arc, it will bend towards the normal. The radius \( r \) of the arc will determine how sharply the light bends.

For light to pass through the arc, the angle of incidence at the point where the light exits the arc must be less than the critical angle. The relationship between the radius of curvature \( r \) and the angle of incidence can be derived from the geometry of the situation.

Using Geometry to Find the Minimum Radius

When light enters the arc, it strikes the surface at an angle. The relationship between the radius \( r \) and the angle \( \theta \) can be approximated using the following relationship:

\( \tan(\theta) = \frac{d}{r} \)

Where \( d \) is the distance from the center of the arc to the point of incidence. For a diameter of 4 cm, the radius of the rod is 2 cm. Therefore, the maximum angle \( \theta \) that allows light to pass through without total internal reflection can be set equal to the critical angle:

\( \tan(41.81^\circ) = \frac{2 \text{ cm}}{r} \)

Calculating the Minimum Radius

Now, we can calculate \( r \):

\( r = \frac{2 \text{ cm}}{\tan(41.81^\circ)} \)

Using a calculator, we find:

\( \tan(41.81^\circ) \approx 0.9 \)

Thus:

\( r \approx \frac{2 \text{ cm}}{0.9} \approx 2.22 \text{ cm} \)

Final Result

The smallest radius \( r \) that permits light to pass through the arc without total internal reflection is approximately 2.22 cm. This means that if the radius of the arc is less than this value, light will be reflected back into the glass rather than passing through into the air.