The following is part of a student’s report on an experiment to measure the focal length of a concave mirror - Leaving Cert Physics - Question 3 - 2013

1. Ensure the measurement is taken from the back of the mirror: Always measure the distance from the pole (center) of the mirror to avoid parallax error.

2. Keep both the screen and mirror vertical: This minimizes distortion in measurements and ensures accuracy.

The student was unable to form an image on the screen when the object was close to the mirror because, at this proximity, the object was within the focal point of the concave mirror. In this range, the rays of light coming from the object diverge rather than converge, resulting in the formation of a virtual image behind the mirror, which cannot be projected onto the screen.

The formula to calculate the focal length (f) of the mirror is given by:

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

Where:

• u = object distance
• v = image distance

Using the provided data, calculate 1/f for each pair of values:

1. For u = 24.0 cm, v = 72.5 cm: $\frac{1}{f_1} = \frac{1}{24.0} + \frac{1}{72.5} \approx 0.0420 + 0.01379 = 0.05579 \Rightarrow f_1 \approx 17.9 \, \text{cm}$

2. For u = 32.0 cm, v = 40.3 cm: $\frac{1}{f_2} = \frac{1}{32.0} + \frac{1}{40.3} \approx 0.03125 + 0.0248 = 0.05606 \Rightarrow f_2 \approx 17.8 \, \text{cm}$

3. For u = 40.0 cm, v = 27.0 cm: $\frac{1}{f_3} = \frac{1}{40.0} + \frac{1}{27.0} \approx 0.025 + 0.03704 = 0.06204 \Rightarrow f_3 \approx 16.1 \, \text{cm}$

4. For u = 48.0 cm, v = 27.9 cm: $\frac{1}{f_4} = \frac{1}{48.0} + \frac{1}{27.9} \approx 0.020833 + 0.03594 = 0.056773 \Rightarrow f_4 \approx 17.6 \, \text{cm}$

Calculate the average focal length: $f_{avg} = \frac{f_1 + f_2 + f_3 + f_4}{4} = \frac{17.9 + 17.8 + 16.1 + 17.6}{4} = 17.4 \, \text{cm}$

Thus, the focal length of the mirror is approximately 17.4 cm.

Before starting the experiment, the student could have estimated the focal length of the concave mirror using the following method:

1. Use a Distant Object: Position a distant object (like a distant tree or building) in front of the mirror to observe where the image forms on the screen. The distance from the mirror to the screen when a clear image is formed is approximately equal to the focal length.

2. Ray Diagram: Construct a ray diagram for the concave mirror showing the path of light rays. This would also provide a visual representation that helps estimate the focal point.