Q30 of 32 Page 60

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.


Given:


The change in the effective refractive index of the medium given by, r is the distance from the center of mass, G is the universal gravitational constant, M the mass of the body and c the speed of light in vacuum, To find the angle of deviation we need to find the relation between angle of incidence and refractive index in terms of mass, gravity and speed of light.


Formula Used:


Applying Snell’s Law, 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 and the second formula applied is the Gravitational Radiation Shift, the shift occurs when passing through a gravitational field with loss of energy changing the refractive index of the light.



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.



where


G is the universal gravitational constant, M is the mass of the earth, r is the distance from the center of the mass and c is the speed of light = .


Explanation:


Using Snell’s Law,








Putting in place of and and



After integration we get







Therefore, the angle of deviation is. Answer


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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.


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(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.



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bmin < b < bmax


b < bmin


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