A laser beam of wavelength 1 m is split and sent into two vacuum cavities of equal length L as shown in the figure. A detector can register an interference signal only if the phase difference between the returning beams is at least 5 10⁻¹¹ rad. A certain physical effect changes the length of cavity 2 by an amount L such that L/L 10⁻²¹ . The minimum cavity length (in km) needed for measuring this physical effect is ap

Step-by-step solution

The path length of the laser beam in cavity 1 is 2L and in cavity 2 is 2(L + L) . The path difference between the two returning beams is: x = 2(L + L) - 2L = 2 L The corresponding phase difference is given by: = 2 x = 4 L We are given that the minimum detectable phase difference is = 5 10⁻¹¹ rad and the fractional change in length is L L = 10⁻²¹ , which gives L = 10⁻²¹ L . Substituting the given values into the phase difference equation: 5 10⁻¹¹ = 4 10⁻⁶ (10⁻²¹ L) 5 10⁻¹¹ = 4 10⁻¹⁵ L Solving for L : L = 5 10⁻¹¹ 4 1