Determination of the Focal Length of a Concave Mirror

Using source–mirror distance S0 and mirror–screen distance Si

1. Objective

To determine the focal length (f) of a concave mirror using measured values of the source–mirror distance S0 and the mirror–screen distance Si.

2. Theory

When a concave mirror forms a sharp real image of an object on a screen, the distances satisfy the mirror formula:

1/f = 1/S0 + 1/Si

where:

  • S0: distance from the light source (object) to the mirror
  • Si: distance from the mirror to the screen
  • f: focal length of the concave mirror

The magnification of the image is given by:

m = Si / S0

3. Apparatus

  • Concave mirror with stand
  • Illuminated object or object pin
  • Screen
  • Optical bench
  • Meter scale or measuring tape

4. Procedure

  1. Fix the concave mirror vertically on the optical bench.
  2. Place the light source at a known distance S0 from the mirror.
  3. Move the screen along the bench until a clear, sharp, inverted image is formed.
  4. Measure and record:
    • S0 – source–mirror distance (in cm)
    • Si – mirror–screen distance (in cm)
  5. Repeat the steps for at least 5–7 different values of S0.
  6. Use the mirror formula to calculate the focal length f for each trial.
  7. Plot a graph of 1/Si (y-axis) versus 1/S0 (x-axis).
  8. Determine the focal length from the data and from the graph.

5. Interactive Data Table and Graph

Use this interactive table to enter your measured values of S0 and Si. The tool will automatically calculate 1/S0, 1/Si, and the focal length f for each trial, then draw a graph of 1/Si versus 1/S0.

Trial S0 (cm) Si (cm) 1/S0 (cm⁻¹) 1/Si (cm⁻¹) f (cm)

Graph: 1/Si vs 1/S0

6. Analysis and Result

For each row of data, the focal length is calculated from the mirror formula:

f = 1 / (1/S0 + 1/Si)

The program computes the focal length for each valid trial and then finds the average focal length. It also performs a linear regression on the 1/Si vs 1/S0 data to estimate the intercept, which should be close to 1/f.

In a typical setup, the focal length of a small concave mirror may be around 10 cm, but the exact value depends on your measured data.

7. Scientific Interpretation for Students

  • Concave mirrors converge parallel light rays to a focal point in front of the mirror.
  • The relation between S0 and Si is not arbitrary; it follows the mirror formula.
  • The graph of 1/Si versus 1/S0 verifies the geometrical optics model.
  • Understanding focal length is important in designing devices such as headlights, telescopes, and shaving mirrors.
  • Small deviations from the theoretical value of f may be due to measurement errors, alignment issues, or parallax.

8. Notes for Good Practice

  • Ensure the bench scale is read from the same side each time to avoid parallax errors.
  • Adjust the screen slowly near the sharp image position for better accuracy.
  • Repeat measurements and discard outliers that are clearly inconsistent.