Optics 1 Question 11
11. A wire is bent in the shape of a right angled triangle and is placed in front of a concave mirror of focal length $f$ as shown in the figure. Which of the figures shown in the four options qualitatively represent(s) the shape of the image
(c) If Statement I is true; Statement II is false.
of the bent wire? (These figures are not to scale.) (2018 Adv.) (a)
(c)
(d)
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Answer:
Correct Answer: 11. (d)
Solution:
- Image of point $A$
$$ \frac{P Q}{x}=\frac{A B}{f / 2} \Rightarrow P Q=\frac{2(A B) x}{f} $$
For $A$ :
$$ \frac{1}{v}+\frac{1}{[-(f / 2)]}=\frac{1}{-f} \Rightarrow v=f $$
$$ \begin{array}{cc} \Rightarrow & \frac{I _{A B}}{A B}=-\frac{v}{u}=-\frac{f}{-\frac{f}{2}} \\ \Rightarrow & I _{A B}=2 A B \end{array} $$
For height of $P Q$,
$$ \begin{aligned} & \frac{1}{v}+\frac{1}{-[(f-x)]}=\frac{1}{-f} \\ & \Rightarrow \frac{1}{v}=\frac{1}{(f-x)}-\frac{1}{f} \Rightarrow v=\frac{f(f-x)}{x} \\ & \Rightarrow \quad \frac{I _{P Q}}{P Q}=-\frac{v}{u}=\frac{f(f-x)}{x[(f-x)]}=\frac{f}{x} \\ & \Rightarrow \quad I _{P Q}=\frac{f}{x} P Q=\frac{f}{x} \quad \frac{2(A B) x}{f} \quad \because P Q=\frac{2(A B) x}{f} \\ & I _{P Q}=2 A B \end{aligned} $$
(Size of image is independent of $x$. So, final image will be of same height terminating at infinity.)
