Question

The sun (diameter d) subtends an angle $\theta$ radian at the pole of a concave mirror of focal length f. The diameter of the image of sun formed by mirror is

• A. $\theta f$
• B. $\frac{\theta f}{2}$
• C. $\theta d$
• D. $\frac{\theta d}{2}$

Answer 1

The correct option is A $\theta f$


At the heart of question lies the idea that infinity is not a number, it's an idea! When we say that the sun will form an image at the focus, as it is 'infinitely far away', we mean that it is so far away that the rays from it are nearly parallel, so the image will form nearly at the focus!
Now, the diameter of the sun can be estimated in terms of its distance from the concave mirror's pole using the fact that it subtends an angle $\theta$ at the pole and[Equation]by using the approximation that the d is nearly equal to the arc for the angle $\theta$ subtended by the sun, R away. (remember this is called the parallax approximation?)
So, $d=R\theta$
Where R can be approximated as the object distance, so, according to sign convention,u =-R.
Now, the image is formed at the focus, as, for the mirror, the sun acts as an optical object infinitely away from it! So, the image distance, according to sign convention is v = -f.
Now, to find the diameter of the image, let us use the relationship between u, v, $h_0$ and $h_i$;
$\frac{h_i}{h_0}=-\frac{v}{u}$,
or, $\frac{d_1}{d}=-\frac{-f}{-R}$,
or, $\frac{d_i}{R\theta }=-\frac{-f}{-R}$
Therefore, the diameter of the image is $f\theta$