Question

Which of the following mirrors forms an image which is virtual and smaller in size?
(A) Convex
(B) Concave
(C) Plane
(D) All of these

Answer 1

Hint: To answer this question, we have to see which mirror makes the rays meet behind its surface after reflection. For the size of the image, we need to use the magnification formula.

Formula Used
\[\dfrac{1}{f} = \dfrac{1}{v} + \dfrac{1}{u}\], \[f = \]focal length of the mirror, \[v = \] image distance, and \[u = \] object distance
\[m = - \dfrac{v}{u}\], \[m\] is the magnification

Complete step-by-step solution
The mirror equation is given as
\[\dfrac{1}{f} = \dfrac{1}{v} + \dfrac{1}{u}\] (1)
And the magnification formula for a mirror is given by
\[m = - \dfrac{v}{u}\] (2)
Multiplying both sides by \[ - u\] in equation (1), we get
\[\dfrac{{ - u}}{f} = \dfrac{{ - u}}{v} + \dfrac{{ - u}}{u}\]
\[\dfrac{{ - u}}{f} = \dfrac{{ - u}}{v} - 1\]
On rearranging
\[\dfrac{{ - u}}{v} = 1 - \dfrac{u}{f}\]
\[\] (3)
From (2)
\[m = - \dfrac{v}{u}\]
Taking reciprocal
\[\dfrac{{ - u}}{v} = \dfrac{1}{m}\]
Substituting this in (3)
\[\dfrac{1}{m} = \dfrac{{f - u}}{f}\]
Taking reciprocal, we get
\[m = \dfrac{f}{{f - u}}\] (4)
For both concave and convex mirrors, \[u\] is negative, according to the Cartesian sign convention. So the magnification formula in (4) can be given as
\[m = \dfrac{f}{{f + |u|}}\], where \[|u|\] is the absolute value of \[u\]
As the focal length of a convex mirror is positive, the denominator of the above expression is greater than the numerator.
So, \[m < 1\] for a convex mirror
Hence, a convex mirror always produces an image which is smaller in size than the object.
Also, the convex mirror diverges the rays after reflecting them from its surface. So the reflected rays appear to diverge from a point on the far side of the object. Therefore, the image produced is virtual in nature.
So, a convex mirror always produces a virtual image which is smaller in size.
The concave mirror converges the rays after reflecting them from its surface. So the image is formed on the same side of the object. Therefore, the image formed is real.
The plane mirror always forms a virtual image at the far side of the object. But the image produced has the same size as the object.

Hence, the correct answer is option A, convex mirror.

Note: Don’t get confused by the special case of the image formation by the concave mirror. In this case, when the object is placed between the principal focus and the pole of the mirror, the image is produced behind the mirror. So the image is virtual. But, at the same time it is magnified, not diminished.