Q.Prove the laws of refraction using Huygens' principle.

Step 1. A plane wavefront $AB$ in medium (1), perpendicular to incident rays $L,M$, strikes a plane refracting surface $XY$ obliquely and enters medium (2); point $A$ reaches the boundary before point $B$ does.

Step 2. By the time $B$ reaches the boundary at $B'$, $A$'s own secondary wavelet -- now travelling inside medium (2) at speed $v_2$ -- has expanded to $A'$ in time $t=BB'/v_1$ (the same elapsed time it took $B$ to travel $BB'$ at medium (1)'s speed $v_1$); so $AA'=v_2t$ and $BB'=v_1t$, giving $\dfrac{AA'}{BB'}=\dfrac{v_2}{v_1}$. The refracted wavefront is the tangent line $A'B'$.

Step 3. In the right triangle $\triangle ABB'$: $\sin i=BB'/AB'$ (where $i$ is the angle of incidence). In the right triangle $\triangle B'A'A$: $\sin r=AA'/AB'$ (where $r$ is the angle of refraction), since both triangles share the hypotenuse $AB'$.

Step 4. Dividing: $\dfrac{\sin i}{\sin r}=\dfrac{BB'/AB'}{AA'/AB'}=\dfrac{BB'}{AA'}=\dfrac{v_1}{v_2}$ (using Step 2's ratio inverted). …

Construct the refracted wavefront from secondary wavelets travelling at the (different) speed of the second medium, and form the ratio of the incidenc …

• Using the SAME speed for both wavelets, as in the reflection proof, instead of correctly using v1 for one leg and v2 for the other. …