Question

A short linear object of length b lies along the axis the axis of a concave mirror of local length f at a distance u from the pole of the mirror. The size of the image is approximately equal to:
$
  A.{\text{ b}}{\left( {\dfrac{{u - f}}{f}} \right)^{\dfrac{1}{2}}} \\
  B.{\text{ b}}{\left( {\dfrac{b}{{u - f}}} \right)^{\dfrac{1}{2}}} \\
  C.{\text{ b}}\left( {\dfrac{{u - f}}{f}} \right) \\
  D.{\text{ b}}{\left( {\dfrac{f}{{u - f}}} \right)^2} \\
$

Answer 1

Hint: The object is always placed in front of the mirror hence the object distance is taken as negative whereas the centre of curvature and focus lie in the front of the concave mirror so that the radius of curvature and the focal length are taken as negative in the case of concave mirror.

Complete step-by-step answer:
By using sign convention, from mirror equation, we get;
$
  \dfrac{1}{v} - \dfrac{1}{u} = \dfrac{{ - 1}}{f} \\
  or \\
  v = \dfrac{{fu}}{{f - u}} \\
  so,\dfrac{{\partial v}}{{\partial u}} = \dfrac{{\partial \left( {\dfrac{{fu}}{{f - u}}} \right)}}{{\partial x}} \\
  or,dv = \dfrac{{{f^2}}}{{{{(u - f)}^2}}} \times du \\
  now,du = b(given). \\
  so,dv = \dfrac{{b{f^2}}}{{{{(u - f)}^2}}} = image{\text{ size}}{\text{.}} \\
 $
Therefore option (D) is the correct answer.

Concave mirrors are converging in nature which means that image can be real or virtual depending upon the position of the object. A concave mirror image can be diminished, same size or magnified but if we talk about convex mirrors the image is diminished only. Concave mirror is capable of producing the real images and it is only possible when the object is located at a distance greater than the focal length from the mirror’s surface. A concave mirror is used by a dentist because it gives the dentist a magnified reflection of the mouth while also refracting a bit of light which means the image on the mirror is larger and brighter.

Note: Concave mirrors are made by silvering the outer surface of a part of the hollow glass sphere. The best understood examples of concave mirror are; concave mirror is used in automobiles headlights, torch lights, reflecting telescopes, etc.