Question

What must be the radius of curvature of a concave mirror to form an image of the Sun $1$ inch in diameter. The diameter of the sun is $\dfrac{1}{{100}}$ its distance from earth.

Answer 1

First we need to form a relation between distance of the sun from the mirror, distance of the image from the mirror with the relation given in the question. Then using the mirror formula to find the focal length of the mirror. Now using the focal length and radius of the curvature formula we can solve this problem. Where $\left( {R = 2f} \right)$.

Complete step by step answer:
As per the problem we have to find the radius of curvature of a concave mirror that forms an image of the Sun which is $1$ inch in diameter.
The given condition is that,
The diameter of sun = $\dfrac{1}{{100}}$ its distance from the Earth
The diameter of sun = $\dfrac{1}{{100}} \times u$
Let us assume the focal length of the concave mirror be = f
Distance to sun from earth (Mirror) = u
Distance to the image of the sun from the mirror = v.
We know the mirror formula using the above parameter as,
$\dfrac{1}{f} = \dfrac{1}{u} + \dfrac{1}{v}$
We also know,
$\dfrac{u}{v} = \dfrac{{{\text{diameter}}\,\,{\text{of}}\,\,{\text{Sun}}}}{{{\text{diameter}}\,\,{\text{of}}\,\,{\text{the}}\,\,{\text{image}}}}$
Now on putting the value we will get,
$\dfrac{u}{v} = \dfrac{{\dfrac{u}{{100}}}}{{\text{1}}}$
Hence,
$\dfrac{u}{v} = \dfrac{u}{{100}}$
Now using mirror formula we will get,
$\dfrac{1}{f} = \dfrac{1}{u} + \dfrac{1}{v}$
Multiplying $u$ on both the sides we will get,
$\dfrac{u}{f} = \dfrac{u}{u} + \dfrac{u}{v}$
$\dfrac{u}{f} = 1 + \dfrac{u}{v}$
Now we know the value $\dfrac{u}{v} = \dfrac{u}{{100}}$,
$\dfrac{u}{f} = 1 + \dfrac{u}{{100}}$
Taking RHS $u$ to other side we will get,
$\dfrac{1}{f} = \dfrac{1}{u} + \dfrac{1}{{100}}$
As $u$ is measured in inches and the value is very large then we can neglect $\dfrac{1}{u}$.
Hence the focal length of the concave mirror is equals to,
$f = 100\,inches$
Now we know the,
Radius of curvature is two time that of its focal length of a mirror
$R = 2f$
Hence,
$R = 2 \times 100\,inches = 200\,inches$.

Note: We neglected the $\dfrac{1}{u}$ value in the above equation because $u$ is very large and when larger values are inverse then it becomes very small that it is neglected in this type of equation. Remember that radius of curvature of a concave mirror is the radius of an imaginary hollow sphere of which concave mirror is a part and the focal length is defined as the distance of the point from the center at which all the parallel rays are covered after reflection.