Determination of the Focal Length of a Concave Mirror

An object pin is placed in front of a mounted concave mirror so that its tip lies along the axis of the mirror.
The position of the pin is adjusted until it coincides with its image such that there is no parallax between the
pin and its image. The distance r of the pin from the mirror is measured.
The required focal length

NOTE
(i) In the position where there is no parallax between the object pin and its image, there is no relative motion between the object and its image when the observer moves the head from side to side.
(ii) When the pin coincides with its image, the rays are incident normal to the mirror and are thus reflected along their own path. Therefore the pin coincides with its image at the Centre of curvature of the mirror.

An illuminated bulb, a screen and a concave mirror mounted in a holder are aligned as shown above.
The mirror position is adjusted to or from the screen until a sharp image of the cross-wire is formed on the screen
besides the object.
The distance r of the mirror from the screen is measured.
The required focal length

Method (3) Using no parallax method in locating V

An object pin P1 is placed at a distance u in front of a mounted concave mirror so that its tip lies along the axis of the mirror.
A search pin P2 placed between the mirror and pin p1 is adjusted until it coincides with the image of pin p1 by no-parallax method.
The distance v of pin p2 from the mirror is measured.
The procedure is repeated for several values of u and the results are tabulated including values of uv, and
u+v.
A graph of uv against u+v is plotted and the slope s of such a graph is equal to the focal length f of the mirror.

Determination of Focal Length of a Convex Mirror

Method (1) Using a convex lens

An object O is placed in front of a convex lens to form a real image on the screen at I1. The distance LI1 of the screen from the lens is then measured.
A convex mirror is placed between the lens and the screen and the mirror is then moved along the axis
OI1 until an image I2 is formed besides O.
The distance LP is measured. The required focal length

NOTE;
When the incident rays from an object are reflected back along the incident path, a real inverted image is formed
besides the object in which case the rays strike the mirror normally.

Therefore they will if produced pass through the Centre of curvature of the mirror thus distance PI1 = radius of curvature.

Method (2) Using No parallax

An object pin O is placed in front of a convex mirror and a virtual diminished image is formed at I.

A plane mirror M is placed between O and P so as to intercept half the field of view of the convex mirror.
Mirror M is adjusted until its own image of O coincides with I by no parallax method. Measure the distances x and y.
The focal length of the mirror is the calculated from
Note:
(i) The two images coincides when they are as far behind the plane mirror as the object is in front.
(ii) Substituting for u = x  y and v = y  x in the mirror formula gives

EXAMPLES
1. An object O is placed 40cm in front of a convex lens of focal length 15cm forming
an image on the screen. A convex mirror situated 4cm from the lens in the region
between the lens and the screen forms the final image besides object O.
(i ) Draw a ray diagram to show how the final image is formed.
(ii) Determine the focal length of the convex mirror.

A plane mirror is placed 10cm in front of a convex mirror so that it covers about half of the mirror surface. A pin
20cm in front of the plane mirror gives an image in it, which coincides with that of the pin in the convex mirror. Find
the focal length of the convex mirror.

EXERCISE
1. Describe an experiment to determine the focal length of a concave mirror.
2. You are provided with the following pieces of apparatus: A screen with cross wires, a lamp, a concave mirror,
and a meter ruler. Describe an experiment to determine the focal length of a concave mirror using the above
apparatus.
3. Describe an experiment, including a graphical analysis of the results to determine the focal length of a concave
mirror using a no parallax method.
4. Describe an experiment to measure the focal length of a convex mirror
5. Describe how the focal length of a diverging mirror can be determined using a convex lens.
6. Describe how the focal length of a convex mirror can be obtained using a plane mirror and the no parallax
method.
7. A plane mirror is placed at a distance d in front of a convex mirror of focal length f such that it covers about half
of the mirror surface. A pin placed at a distance L in front of the plane mirror gives an image in it, which coincides
with that of the pin in the convex mirror. With the aid of an illustration, Show that 2df = d2  L2