Question

The image of an object $O$ due to reflection from the surface of a lake is elongated due to the ripple s on the water surface caused by a light breeze. This is because the ripples act as tilted mirrors as shown below. Consider the case, where $O$ and the observer $E$ are at the same height above the surface of the lake. If the maximum angle that the ripples make with the horizontal is a , the $n$ the angular extent $\delta$ of the image will be

• A. $\frac{\alpha }{2}$
• B. $\alpha$
• C. $2\alpha$
• D. $4\alpha$

Answer 1

Answer: (C)

We can consider ripples as plane mirrors reflecting light to form image with angular extent $\delta$.
It is given $O$ and $E$ are at same level, so $OEBC$ is a trapezium with $OE||BC$.
Now, different angles of trapezium $OEBC$ are as shown below.

$\angle 1=90^{\circ }-\beta -\alpha$
$\angle 2=90^{\circ }-\gamma -\alpha$
$\angle OPB=\angle EPC$ and $\angle OPE=\angle BOC$
$\Rightarrow 90^{\circ }-\beta -\alpha =90^{\circ }-\gamma -\alpha$
$\Rightarrow \beta =\gamma$ ...(i)
As , $\angle OPC=180^{\circ }$
$\Rightarrow 180^{\circ }=\beta +\gamma +2\alpha +180-2\gamma -\delta$
$\Rightarrow 2\alpha +\beta -\gamma -\delta =0$
$\Rightarrow \delta =2\alpha [\because \beta =r$ from Eq. (i)$]$