Focal length of a concave mirror by finding v for different u (Class 12 practical)

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

• •Optical bench with three uprights and clamps
• •Concave mirror (focal length less than 20 cm)
• •Two sharp-edged pins (object and image pins)
• •Index needle (a knitting needle will do)
• •Metre scale
• •Spirit level

Principle

Concave mirror: u-v method for focal length: formulas and key facts

Mirror formula

$\frac{1}{v}+\frac{1}{u}=\frac{1}{f}$

Cartesian sign convention: distances measured from the pole, positive in the direction of the incident light.

Focal length from magnitudes

$f=\frac{uv}{u+v}$

For a real image of a real object with a concave mirror.

Index correction

$e=\text{actual distance}-\text{observed distance};u=u_{obs}+e$

Focal length and radius of curvature

$f=\frac{R}{2}$

Graph methods

$\text{uv against }(u+v):\text{slope}=f;\frac{1}{v}\text{ against }\frac{1}{u}:\text{intercepts}=\frac{1}{f}$

Key facts

• •Object between F and 2F: image beyond 2F, real, inverted, magnified
• •Object at 2F: image at 2F, same size
• •No parallax: image and pin move together
• •Tip of the image touches the tip of the pin, not overlapping
• •Find f row by row, then average; never from mean u and mean v

Procedure

1. 1Find a rough focal length by focusing the image of a distant tree or building on a wall or sheet of paper and measuring the mirror-to-image distance.
2. 2Place the optical bench on a rigid table and level it with a spirit level.
3. 3Mount the mirror vertically near one end. Move the object pin along the bench and adjust the pin height or mirror tilt until the image stays at the same height, so the principal axis is parallel to the bench.
4. 4Adjust both pins so their tips are at the height of the pole of the mirror.
5. 5Find the index correction for each pin: touch one end of the index needle to the pin tip and the other to the pole, read the uprights, and compare the observed distance with the needle's length measured on a metre scale.
6. 6Place the object pin between F and 2F. Look along the bench for its inverted image. Move the image pin until there is no parallax between its tip and the tip of the image.
7. 7Note the positions of the mirror, object pin and image pin uprights, and so the observed u and v.
8. 8Repeat for at least five more positions of the object pin.
9. 9Add the index correction to each observed u and v, find f for each row, and take the mean.

Observation table

Draw the table like this in your record. The numbers are sample readings to show how it is filled in; use your own.

Calculation

1. 1.Row 3: $f=\frac{uv}{u+v}=\frac{24.0\times 40.3}{24.0+40.3}=\frac{967.2}{64.3}=15.04\approx 15.0$ cm
2. 2.Row 2: $f=\frac{22.0\times 46.6}{68.6}=\frac{1025.2}{68.6}=14.94\approx 14.9$ cm
3. 3.Check with signs, row 3: $u=-24.0$ cm, $v=-40.3$ cm; $\frac{1}{f}=\frac{1}{-40.3}+\frac{1}{-24.0}=-\frac{64.3}{967.2}$, so $f=-15.0$ cm (negative: a concave mirror).
4. 4.Mean focal length of the six rows $=\frac{15.0+14.9+15.0+14.9+15.1+14.9}{6}=14.97\approx 15.0$ cm.

Result

The focal length of the given concave mirror is f = ____ cm (sample: 15.0 cm). In the Cartesian sign convention it is written as −15.0 cm.

Precautions

• →Keep the uprights rigid and vertical, and the bench level.
• →Keep the object pin between F and 2F, so the image is real, magnified and easy to locate.
• →Use a mirror of small aperture so the image is sharp.
• →Keep your eye about 25 cm from the image pin while removing parallax.
• →The tip of the image must just touch the tip of the image pin, not overlap it.
• →Do not interchange the object and image pins during the experiment.
• →Calculate f for each row from corrected u and v, then average; do not use the mean of u and the mean of v.
• →Never look at the image of the Sun.

Sources of error

• •The bench top may not be horizontal, or the pin tips and the pole may not be at the same height.
• •If the mirror is not front-coated, reflection from the glass surface gives a second faint image.
• •Parallax may not be removed completely.
• •The index correction may be read inaccurately.

Viva questions with answers

Say your answer out loud first, then open the card. Examiners want one or two clear sentences, not a paragraph.

1. 1.Define the focal length of a concave mirror.Show answer

   The distance between the pole and the principal focus, where rays parallel to the principal axis meet after reflection.

2. 2.State the mirror formula.Show answer

   $1/v+1/u=1/f$, with distances measured from the pole using the Cartesian sign convention.

3. 3.What is parallax, and how do you remove it?Show answer

   The apparent relative movement of two objects when the eye moves sideways. Move the image pin until it and the image move together as you move your eye; then they are at the same place.

4. 4.What is index correction?Show answer

   The difference between the actual distance (length of the index needle) and the observed distance between the uprights. It is added to every observed u and v.

5. 5.Why is the object placed between F and 2F?Show answer

   The image is then real, inverted, magnified and beyond 2F, so it is bright and easy to locate with a pin.

6. 6.Can you find f this way if the object is placed between F and the pole?Show answer

   No. The image is virtual and behind the mirror, so it cannot be located on the bench with a pin.

7. 7.Where is the image when the object is at 2F (the centre of curvature)?Show answer

   At 2F itself: real, inverted and the same size as the object. Then u = v = 2f.

8. 8.What is the relation between focal length and radius of curvature?Show answer

   $f=R/2$, for mirrors of small aperture.

9. 9.Does the focal length of a mirror change if it is immersed in water?Show answer

   No. A mirror works by reflection, and the laws of reflection do not depend on the medium.

10. 10.Why is the focal length of a concave mirror negative?Show answer

    In the Cartesian sign convention the focus lies in front of the mirror, opposite to the direction of the incident light, so f is negative.

11. 11.Why does a dentist use a concave mirror?Show answer

    With the tooth closer than F, the mirror gives an erect, magnified virtual image, so the tooth is seen larger.

12. 12.How can you find f from a graph?Show answer

    Plot uv against u + v: the slope is f. Or plot 1/v against 1/u: both intercepts equal 1/f.

Viva practice: these questions one at a time, with the answer hidden until you ask, and a tally of what to revise.

Common mistakes in the practical exam

The slips examiners see most often, and what to write instead.

• ✗Mistake: Using the mean u and mean v to find f.

  NCERT's precautions say f must be found row by row and then averaged.

  ✓Do this: Work out f in every row, then take the mean of the f column.

• ✗Mistake: Forgetting the index correction or subtracting it.

  ✓Do this: Write e = actual − observed (with its sign) and add e to every observed u and v.

• ✗Mistake: Overlapping the image pin and the image instead of touching tips.

  ✓Do this: Bring the image pin up until the two tips just meet and move together.

• ✗Mistake: Mixing signs: putting u negative and v positive for a real image.

  ✓Do this: Either use magnitudes in f = uv/(u + v), or put both u and v negative in the mirror formula.

• ✗Mistake: Drawing the ray diagram with the object beyond 2F although readings were taken between F and 2F.

  ✓Do this: Draw the ray diagram for the case you actually used.

Concave mirror: u-v method for focal length: frequently asked questions