Question

A cube of side $2\,m$ is placed in front of a concave mirror of focal length $1\,m$ with its face P at a distance of $3\,m$ and face B at a distance of $5\,m$ form the mirror. The distance between the images of faces A and B and heights of images of A and B are,respectively

A. $1{\text{ }}m,\,0.5{\text{ }}m,\,0.25{\text{ }}m$
B. $0.5{\text{ }}m,\,1\,m,\,0.25{\text{ }}m$
C. $0.5\,m,\,0.25\,m,\,1\,m$
D. $0.25\,m,\,1\,m,\,0.5\,m$

Answer 1

First using the mirror equation to find the distance between the two points ${v_A}$ and ${v_B}$ then use the magnification relation to find the height of the images for point A and point B.
The formula for the mirror equation is
$\dfrac{1}{f} = \dfrac{1}{u} + \dfrac{v}{f}$
Here we have to also use the magnification relation. Lens magnification is known as the ratio of an image’s height to an object’s height.
The formula for magnification is $\dfrac{{{\text{Image}}\,{\text{height}}}}{{{\text{Object}}\,{\text{height}}}}{\text{ = -}}\dfrac{{{\text{Image}}\,{\text{distance}}}}{{{\text{Object}}\,{\text{distance}}}}$

Complete step by step answer:
Here,
By using the mirror equation,
$\dfrac{1}{f} = \dfrac{1}{u} + \dfrac{v}{f}$
Here,
For point A,
$\dfrac{1}{{{v_A}}} = - 1 + \dfrac{1}{3} = - \dfrac{2}{3}$
So,
${v_A} = - \dfrac{3}{2} = - 1.5$
Again,
For point B,
$\dfrac{1}{{{v_B}}} = - 1 + \dfrac{1}{5} = - \dfrac{4}{5}$
So,
${v_B} = - \dfrac{5}{4} = - 1.25$
Therefore,
The distance between two points is $= 1.5 - 1.25 = 0.25\,m$
Now,
Using the magnification relation, we know that,

{{{\text{Object}}\,{\text{height}}}}{\text{ = - }}\dfrac{{{\text{Image}}\,{\text{distance}}}} {{{\text{Object}}\,{\text{distance}}}}$$ …… (1) Therefore, For point A, Image height $$ = - 2 \times \dfrac{{ - 1.5}}{{ - 3}} = - 1\,m{\text{ = 1}}\,m$$ …… (Using equation (1)) And, For point B, Image height $$ = - 2 \times \dfrac{{ - 1.25}}{{ - 5}} = - 0.5\,m{\text{ = 0}}{\text{.5}}\,m$$ …… (Using equation (1)) Hence, The distance between two points is $${\text{0}}{\text{.25}}\,m$$ Image height for point A is $${\text{1}}\,m$$ Image height for point B is $${\text{0}}{\text{.5}}\,m$$ **Note:** An optical system's focal length is a measure of how intensely the system converges or diverges light; it's the opposite of the optical strength of the system. A positive focal length means that light is converged by a system, while a negative focal length implies light is diverged by the system.A curved mirror is a mirror with a curved surface which reflects. The surface can be convex or concave. Many curved mirrors have surfaces shaped like part of a sphere but often other shapes are used in optical devices. Magnification is the method of enlarging something's apparent size, not its physical size.