Animated Solution for Physics - Optics: In the adjacent diagram, $CP$ represents a wavefront and $AO$ and $BP$, the corresponding two rays. Find the condition of $\theta$ for constructive interference at $P$ between the ray $BP$ and reflected ray $OP$

Imagine a plane wavefront $CP$ advancing towards a horizontal mirror. From this wavefront, we track two specific rays that eventually meet at point $P$. The first ray travels directly from $B$ to $P$. The second ray takes a slightly more adventurous route: it travels from $C$ to $O$, strikes the mirror, and reflects towards $P$.

Because $CP$ is a wavefront, all points on it are perfectly in phase. This means the light at $C$ and the light at $P$ (on the direct ray) start their journey to the interference point with the exact same phase. The geometric path difference between these two rays is simply the extra distance the reflected ray has to travel.

The geometric path difference $\Delta x$ is the sum of the distances $CO$ and $OP$:

However, we must not forget a crucial physical phenomenon. When light reflects off a denser medium (like our mirror), it undergoes an abrupt phase shift of $\pi$ radians. In terms of path length, this is equivalent to an additional shift of $\lambda /2$.

For constructive interference, the total phase difference must be a multiple of $2\pi$. Because the reflection already provides a $\pi$ shift, the geometric path difference must provide the remaining odd multiple of $\pi$. Therefore, the condition for constructive interference becomes:

Let's use some trigonometry to find $CO$ and $OP$. In the right-angled $△ORP$, where $OR=d$ is the vertical distance from the mirror to $P$:

Next, look at the right-angled $△COP$. The angle between the incident ray $CO$ and the reflected ray $OP$ is $2\theta$. Therefore:

Now, substitute these into our path difference equation:

Using the double-angle identity $1+\cos 2\theta =2{\cos }^2\theta$, we can simplify this beautifully:

Finally, equating this to the condition for the first maximum ($n=1$):

This elegant result perfectly matches option (b).