To find the focal length of a Concave Mirror – Experiment

To find the focal length of a Concave Mirror – Experiment.

Aim

To find the value of v for different values of u in case of a concave mirror and to find the focal length.

Apparatus and Material Required

• An optical bench,
• two sharp-edged needles (pins),
• concave mirror of less than 20 cm focal length,
• three uprights (with clamps),
• index needle (may be a knitting needle),
• metre scale and spirit level.

Terms and Definitions

• Principal Axis is the line passing through the center of curvature and pole of the mirror.
• Principal Focus (F) is the point where rays parallel to the principal axis of the concave mirror, converge after reflection from the surface of the mirror.
• Pole (P) is the geometrical center of the mirror.
• Focal Length is the distance between the pole, P and focus, F.

Cartesian Sign Convention

• The object is always placed to the left of the mirror.
• All the distances are measured from the pole \(P\) of the mirror.
• The distance measured in the direction of propagation of incident rays is positive while those measured in the opposite direction is negative.
• The height measured upwards is positive and measured downwards is negative.

Principle

If an object is at a distance \(u\) from the pole of a concave mirror of focal length \(f\) and its image forms at a distance \(v\) from the pole, then by the mirror formula \[\frac{1}{f}=\frac{1}{u}+\frac{1}{v}\]

\[\text{or }f = \frac{uv}{u+v}\]

The image of an object placed at a position lying in between the prinicpal focus, F and the center of curvature, C is real, inverted and magnified. The image positions is in between the center of curvature of the mirror and the infinity. Refer the adjoining diagram.

Procedure

1. Obtain approximate value of the focal length of concave mirror by focusing the image of a distant object. It is the distance between the mirror and the screen on which a bright and clear image of distant object forms.
2. Make the optical bench horizontal by using spirit level and levelling screws.
3. Clamp the concave mirror on an upright and mount it vertically near one end of the optical bench. Move an object pin \(P_1\) on the optical bench back and forth so that its image is formed at the same height.
4. Place another vertically mounted sharp and bright pin \(P_2\) in front of the reflecting surface of the concave mirror.
5. Move the pin \(P_1\) away from the mirror and place it almost at a distance of twice of focal length (roughly estimated from step 1). Make suitable movement of the pin so that an inverted image of same size is visible.
6. Place a piece of paper on the tip of one pin \(P_1\). Consider it as object pin. Place the pin with the paper at a distance lying between \(F\) and \(2F\).
7. Locate the image of the pin with paper using the other pin.
8. Note down the values of \(u\) and \(v\) i.e. the distances of the object and the image pins from the mirror respectively.
9. Repeat the steps 6 to 8 at least five times to get five set of the values of \(u\) and \(v\).

Observations

Rough focal length of the concave mirror =10 cm.

Calculations

Mirror formula, \(\frac{1}{f} = \frac{1}{u} + \frac{1}{v}\) \[f = \frac{uv}{u+v}\]

Mean focal length, \(f_{mean} = \frac{10.1+10.2+10.1+10.2+10.3}{5}\) = 10.2 \(cm\)

Results

The focal length of the given concave (converging) mirror is 10.2 cm.

Precautions

• The uprights supporting the optical elements should be rigid and mounted vertically.
• The object pin should be kept in between the centre of curvature and the focus of the mirror.
• The aperture of mirror should be small otherwise the image formed will not be distinct.
• Eye should be placed at a distance of distinct vision (25 cm) from the image needle.
• The tip of the inverted image of the object pin must touch the tip of the image pin and must not overlap.
• The image and the object pins should not be interchanged during the course of the experiment.
• A white screen or plane background may be used for seeing the clear image of the pin.
• Image of the Sun should not be seen directly as it may hurt your eyes.

Sources of errors

• The top of the optical bench is not horizontal.
• The tips of pins and pole of the mirror are not at the same horizontal level.

Conclusion

When an object is placed between the focus and center of curvature of a concave mirror, it forms a real, inverted, and magnified image, with the image distance increasing as the object nears the focus. The focal length is found using mirror formula.

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Reference: NCERT Lab Manual