Question

In the experiment to determine the focal length of a concave mirror by graphical methods, the u-v graph is a ___________.
A) straight line
B) curve
C) circle
D) ellipse

Answer 1

For a concave mirror, the object whose image is to be observed is always placed to the left of the mirror, and therefore the distance of the object from the mirror is always negative. The principal focus of the concave mirror is also located to the left of the concave mirror.

Formula used:
The mirror equation is given by, $\dfrac{1}{v} + \dfrac{1}{u} = \dfrac{1}{f}$ where $v$ is the distance from the mirror where the image is formed, $u$ is the distance at which the object is placed and $f$ is the focal length of the mirror.

Complete step by step answer:
Step 1: Explain the experiment to determine the focal length of the mirror.
To determine the focal length of a concave mirror, the experimenter places an object on the principal axis of the concave mirror at some distance from the mirror. This distance is termed as the object distance $u$. The experimenter then observes the image of the object on a screen placed in front of the mirror at some distance. This distance is termed as the image distance. Then using the mirror equation the focal length can be determined.
The setup is given below.
Step 2: Express the mirror equation of the concave mirror to obtain an expression for the image distance.
The mirror equation is given by, $\dfrac{1}{v} + \dfrac{1}{u} = \dfrac{1}{f}$ where $v$ is the distance from the mirror where the image is formed, $u$ is the distance at which the object is placed and $f$ is the focal length of the mirror.
Then in terms of the image distance, $\dfrac{1}{v} = \dfrac{1}{f} - \dfrac{1}{u}$
For a concave mirror, $u$ and $f$ are negative.
Then the image distance will be
$\Rightarrow \dfrac{1}{v} = \dfrac{1}{u} - \dfrac{1}{f}$ ------- (1) or, $v = \dfrac{{uf}}{{f - u}} = \dfrac{{ - uf}}{{u - f}}$
Thus the image distance is obtained to be $v = \dfrac{{ - uf}}{{u - f}}$ and hence is also negative.
Step 3: Based on the obtained expression of the image distance, sketch the u-v graph.
As the object distance and the image distance are both negative, the object distance $u$ is taken along the negative x-direction and the image distance $v$ is taken along the negative y-direction.
The equation (1) gives the image distance of the concave mirror as
$\Rightarrow \dfrac{1}{v} = \dfrac{1}{u} - \dfrac{1}{f}$ . This equation suggests that the shape of the graph is a curve.
The $u$ - $v$ graph for the concave mirror is sketched below.

Hence the correct option is B.

Note:
The focal length of the mirror can also be determined by focussing a distant object. The inverted image of the object will then be obtained on a screen placed in front of the concave mirror. The distance between the screen and the mirror will give the focal length of the concave mirror. This is because the image of an object at infinity is formed at the focus of the image.