Focal length of a convex lens by plotting u-v or 1/u-1/v graphs (Class 12 practical)

Aim

To find the focal length of a convex lens by plotting graphs between u and v or between 1/u and 1/v.

Apparatus and material required

• •Optical bench with three uprights and clamps
• •Convex lens (focal length less than 20 cm)
• •Two sharp-edged pins (object and image pins)
• •Index needle
• •Metre scale
• •Spirit level

Principle

Convex lens: focal length from u-v and 1/u-1/v graphs: formulas and key facts

Lens formula

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

Cartesian sign convention.

Focal length from magnitudes

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

u-v graph

$u=v=2f\text{ where the 45° line meets the curve}$

1/u-1/v graph

$\text{x-intercept}=\text{y-intercept}=\frac{1}{f}$

Index correction for a lens

$e=\left(L_0+\frac{t}{2}\right)-\text{observed length}$

L₀: length of index needle on the metre scale; t: thickness of the lens.

Key facts

$Objectbetweenfand2f:imagebeyond2f,real,inverted,magnified|Objectat2f:imageat2f,samesize|u-vgraphisahyperbola|1/u-1/vgraphisastraightlineofslope-1|Samescaleonbothaxes|PowerP=1/f(finmetres),unitdioptre$

Procedure

1. 1Find a rough focal length by focusing a distant tree on a wall or sheet of paper and measuring the lens-to-image distance.
2. 2Level the optical bench with a spirit level.
3. 3Mount the lens vertically near the middle of the bench, with its principal axis parallel to the bench.
4. 4Find the index correction: touch one end of the index needle to the lens surface and the other to the pin tip; compare the observed distance between uprights with the needle length plus half the lens thickness. Do this for both pins.
5. 5Adjust the tips of both pins to the height of the optical centre. Mark the image pin with a small piece of paper.
6. 6Place the object pin a little less than 2f from the lens. Look through the lens from the other side and move the image pin until there is no parallax between it and the inverted image.
7. 7Record the positions of the lens, object pin and image pin uprights.
8. 8Move the object pin 2 cm to 3 cm closer to the lens each time and take at least six sets of readings with the object between f and 2f.
9. 9Apply the index corrections, find 1/u and 1/v, and plot the u-v and 1/u-1/v graphs on the same scale for both axes.

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: $f=\frac{18.0\times 35.6}{18.0+35.6}=\frac{640.8}{53.6}=11.96\approx 12.0$ cm; mean of the six rows $=\frac{11.9+12.1+12.0+12.1+11.9+11.9}{6}=11.98\approx 12.0$ cm.
2. 2.u-v graph: plot u on the x-axis and v on the y-axis (same scale), using each pair both ways. Draw the line u = v from the origin; it meets the smooth curve at about (23.9, 23.9), between the points (24.0, 23.8) and (23.8, 24.0). So 2f = 23.9 cm and f = 11.95 ≈ 12.0 cm.
3. 3.1/u-1/v graph: plot 1/u on the x-axis and 1/v on the y-axis. Draw the best straight line (slope −1); it cuts both axes at about 0.0835 cm⁻¹, the mean of 1/u + 1/v over the six rows (row by row, from 0.0827 to 0.0841).
4. 4.$f=\frac{1}{0.0835}=11.98\approx 12.0$ cm from each intercept.

Result

The focal length of the given convex lens: (i) from the table, f = ____ cm; (ii) from the u-v graph, f = ____ cm; (iii) from the 1/u-1/v graph, f = ____ cm (sample: 12.0 cm each).

Precautions

• →Keep the uprights rigid and vertical, and the bench level.
• →Use a lens of small aperture so the image is sharp.
• →Keep your eye more than 25 cm from the image pin.
• →Keep the pin tips and the optical centre at the same height.
• →Do not interchange the object and image pins, as their index corrections may differ.
• →The tip of the inverted image must just touch the tip of the image pin.
• →Use the same scale on both axes of each graph.

Sources of error

• •The bench may not be horizontal, or the pin tips and optical centre may not be at the same height.
• •Parallax may not be removed completely.
• •The lens may not be thin, so the measured distances from its surface differ from those from the optical centre.
• •Errors in the index correction or in reading the scale.

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 convex lens.Show answer

   The distance between the optical centre and the principal focus, where rays parallel to the principal axis meet after refraction.

2. 2.What is the optical centre of a lens?Show answer

   The point on the principal axis through which a ray passes without deviation.

3. 3.State the lens formula.Show answer

   $1/v-1/u=1/f$, with distances from the optical centre in the Cartesian sign convention.

4. 4.What is the shape of the u-v graph and how do you get f from it?Show answer

   A rectangular hyperbola. The 45° line through the origin cuts it where u = v = 2f, so f is half that value.

5. 5.What is the shape of the 1/u-1/v graph and what are its intercepts?Show answer

   A straight line with slope −1. Both intercepts are equal to 1/f.

6. 6.Why is half the thickness of the lens added in the index correction?Show answer

   Distances are measured from the optical centre, which for an equiconvex lens is at its geometrical centre, half the thickness from the surface touched by the needle.

7. 7.Why is the object kept between f and 2f?Show answer

   The image is then real, magnified and beyond 2f, so it is easy to locate. Moving u from 2f to f covers v from 2f to infinity.

8. 8.Why are u and v said to be interchangeable?Show answer

   Object and image are conjugate points: if the object is placed at the image position, the image forms at the old object position. So each reading gives two points on the graph.

9. 9.What happens to the focal length of a convex lens when it is immersed in water?Show answer

   It increases, because the refractive index of glass relative to water is smaller than relative to air.

10. 10.Does the focal length depend on the colour of light?Show answer

    Yes. The refractive index is greater for violet than for red, so the focal length is shorter for violet light.

11. 11.Define the power of a lens and give its unit.Show answer

    P = 1/f with f in metres. Its unit is the dioptre (D). A convex lens has positive power.

12. 12.What happens to the image if the lower half of the lens is covered?Show answer

    A complete image still forms, but it is less bright, because fewer rays from each point of the object reach the image.

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: Different scales on the two axes of the u-v graph.

  The 45° line no longer means u = v, so 2f is read wrongly.

  ✓Do this: Use the same scale on both axes and start both at zero.

• ✗Mistake: Plotting only six points on the u-v graph.

  ✓Do this: Plot each pair both ways, (u, v) and (v, u), to get twelve points and a full curve.

• ✗Mistake: Forgetting t/2 in the index correction.

  ✓Do this: Actual distance = needle length + half the lens thickness.

• ✗Mistake: Joining the points of the 1/u-1/v graph dot to dot.

  ✓Do this: Draw the best straight line and extend it to meet both axes.

• ✗Mistake: Reading the intercept as f instead of 1/f.

  ✓Do this: Write the intercept in cm⁻¹ and then take its reciprocal.

Convex lens: focal length from u-v and 1/u-1/v graphs: frequently asked questions