Two plane mirrors are inclined at an angle #theta# . Light ray is incident parallel to one of the mirror.For what value of #theta #ray will retrace it's path after third reflection. ?

For the light ray to retrace its path after the third reflection between two plane mirrors inclined at an angle theta, the condition is that the total angle turned by the ray after three reflections must be a multiple of 360 degrees. This means the sum of the angles of incidence and reflection at each mirror must add up to a multiple of 180 degrees.

Let's denote the angle of incidence and angle of reflection at each mirror as α. Since the light ray is incident parallel to one of the mirrors, the angle of incidence is equal to the angle of reflection for the first mirror. Therefore, for the second mirror, the angle of incidence is also α.

For the first reflection, the angle turned is 180 degrees - 2α. For the second reflection, it's 180 degrees - 2α again. For the third reflection, it's just α since the light ray retraces its path. So, the total angle turned after three reflections is:

[ (180^\circ - 2\alpha) + (180^\circ - 2\alpha) + \alpha = 360^\circ - 2\alpha ]

For the light ray to retrace its path, this total angle must be a multiple of 360 degrees. Therefore, ( 360^\circ - 2\alpha = 360^\circ ) or ( -2\alpha = 0 ), which implies ( \alpha = 0 ).

Since α represents the angle of incidence and reflection, it cannot be negative. Hence, for the light ray to retrace its path after the third reflection, the angle of incidence and reflection at each mirror must be 0 degrees. This occurs when the mirrors are perpendicular to each other (θ = 90 degrees).