Q31 of 32 Page 60

An infinitely long cylinder of radius R is made of an unusual exotic material with refractive index –1 (Fig. 9.7). The cylinder is placed between two planes whose normal are along the y direction. The center of the cylinder O lies along the y-axis. A narrow laser beam is directed along the y direction from the lower plate. The laser source is at a horizontal distance x from the diameter in the y direction. Find the range of x such that light emitted from the lower plane does not reach the upper plane.


The range of x such that light emitted from the lower plane does not reach the upper plane


Given:


The radius of the cylinder = R, the refractive index of the cylinder = -1, placed in a y-axis coordinate with center at zero, the angle of incidence is and the angle of refraction is. The distance at which the cylinder is kept on y-axis is x.


Formula used:


Snell’s Law, is the ratio between the sine value of incidence and refraction with the ratio of refractive index of the mediums through which the light passes.



where


is the refractive index of the medium, is the refractive index of the air. i is the angle of incidence and r is the angle of refraction


Explanation:


The angle at which the rays will not reach the cylinder is from. The refracted ray coming out of the cylinder is. Therefore, the range for the angle of refraction in terms of angle of incidence is







Now the above range of x or where the cylinder should be kept to not get light on the plates is .


More from this chapter

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28

Show that for a material with refractive index light incident at any angle shall be guided along a length perpendicular to the incident face.

 29

The mixture a pure liquid and a solution in a long vertical column (i.e., horizontal dimensions << vertical dimensions) produces diffusion of solute particles and hence a refractive index gradient along the vertical dimension. A ray of light entering the column at right angles to the vertical is deviated from its original path. Find the deviation in travelling a horizontal distance d << h, the height of the column.

 30

If light passes near a massive object, the gravitational interaction causes a bending of the ray. This can be thought of as happening due to a change in the effective refractive index of the medium given by


Where r is the distance of the point of consideration from the center of the mass of the massive body, G is the universal gravitational constant, M the mass of the body and c the speed of light in vacuum. Considering a spherical object find the deviation of the ray from the original path as it grazes the object.


 32

(i) Consider a thin lens placed between a source (S) and an observer (O) (Fig. 9.8). Let the thickness of the lens vary as where b is the vertical distance from the pole. w0 is a constant. Using Fermat’s principle i.e. the time of transit for a ray between the source and observer is an extremum; find the condition that all paraxial rays starting from the source will converge at a point O on the axis. Find the focal length.



(ii) A gravitational lens may be assumed to have a varying width of the form


bmin < b < bmax


b < bmin


Show that an observer will see an image of a point object as a ring about the center of the lens with an angular radius