Phase Differences of Light Waves at Specified Points

To determine the correct relation among the phases $\phi_c, \phi_d, \phi_e$, and $\phi_f$, we use the principles of wavefronts in wave optics:

1. Phase on a Wavefront:
   A wavefront is defined as the locus of all points in the medium having the same phase of oscillation.

   • Points $c$ and $d$ lie on the same incident wavefront, which implies:
     $\phi_c = \phi_d \implies \phi_d - \phi_c = 0$
   • Points $e$ and $f$ lie on the same refracted/propagated wavefront, which implies:
     $\phi_e = \phi_f \implies \phi_f - \phi_e = 0$

2. Evaluation of the Options:

   • Option (A): States that $\phi_c$ cannot be equal to $\phi_d$. Since $c$ and $d$ lie on the same wavefront, $\phi_c = \phi_d$, making this option incorrect.
   • Option (B): States that $\phi_d$ can be equal to $\phi_e$. Since the wave propagates from the wavefront containing $c$ and $d$ to the wavefront containing $e$ and $f$, there is a non-zero optical path difference between the two wavefronts (given that $\phi_c \neq \phi_f$). Thus, $\phi_d \neq \phi_e$, making this option incorrect.
   • Option (C): Considers the difference $(\phi_d - \phi_f)$ and $(\phi_c - \phi_e)$:
     $\phi_d - \phi_f = \phi_c - \phi_e \iff \phi_d - \phi_c = \phi_f - \phi_e$
     Since $\phi_c = \phi_d$ and $\phi_e = \phi_f$, both sides equal:
     $\phi_d - \phi_f = \phi_c - \phi_e$
     Therefore, this relation is identically satisfied and correct.
   • Option (D): States that $(\phi_d - \phi_c)$ is not equal to $(\phi_f - \phi_e)$. Since both quantities are equal to zero, they are equal, making this option incorrect.

Hence, the correct choice is (C).