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Home / Physics / Determination of Focal Length of Concave Mirror: Procedure & Precautions - Testbook

Determination of Focal Length of Concave Mirror: Procedure & Precautions - Testbook

Last Updated on Feb 05, 2025
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Purpose of the Experiment

The primary aim of this experiment is to discover the value of v for a range of u values in the case of a concave mirror and to ascertain the mirror's focal length.

Required Materials

Here is a list of the necessary materials for conducting this experiment:

  • An optical bench with three uprights
  • A concave mirror
  • A mirror holder
  • Two optical needles
  • A knitting needle
  • A half-metre scale

Theoretical Background

To determine the focal length, we employ the mirror formula. The mirror formula is represented as:

From the aforementioned equation, the focal length can be deduced as:

Where:

  • f represents the focal length of a concave mirror
  • u is the distance between the object needle and the mirror's pole
  • v is the distance of the image needle from the mirror's pole

According to the sign-convention, u and v possess negative values, hence, the value of f will also be negative.

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Ray Diagram

Procedural Steps

Estimating the Rough Focal Length:

  1. Position the concave mirror in the mirror holder.
  2. Take the mirror holder outside and aim the mirror at a distant structure.
  3. Move the mirror back and forth until a clear image of the structure is visible on a white-painted wall.
  4. The rough focal length of the mirror is the distance between the mirror and the wall.

Setting Up the Mirror:

  1. The fixed upright should be placed near the zero end of the optical bench, with the other two uprights near the opposite end.
  2. Attach the mirror to the holder at the fixed upright such that the mirror faces the other end of the bench and its surface is perpendicular to the length.
  3. Use levelling screws to ensure the mirror surface is vertical.
  4. Note the position of the index mark at the base of the mirror upright.

Setting Up the Object Needle:

  1. Mount a thin optical needle, labelled O, on the second upright.
  2. Move the object needle along with the clamp 1.5 times the rough focal length of the mirror.
  3. Adjust the height of the object needle to align the object needle and the mirror's pole horizontally.
  4. Close your left eye and open your right eye to see an inverted, enlarged image of the needle with the tip of the image at the mirror's centre.
  5. Record the position of the index mark at the base of the object needle.

Setting Up the Image Needle:

  1. Attach a thick optical needle to the third upright and shift it to the other end of the optical bench.
  2. Close your left eye and open your right eye to adjust the height of the image needle so the tip of the image needle and the tip of the image align.
  3. Move your eye to the right to check for parallax. If the tips appear separated, parallax is present.
  4. Remove the parallax tip-to-tip.
  5. Record the index mark at the base of the image needle upright.
  6. Record all index marks as observation 2 to determine index corrections.
  7. Determine the index correction for the distance between the mirror's pole, the tip of the object needle, and the image needle.
  8. Move the object needle upright towards the mirror by 1 cm to obtain observations 2 and 1.
  9. To obtain observations 4, 5, and 6, move the object needle away from the mirror by 1 cm.
  10. Note all the observations.

Observations

The rough focal length of the given concave mirror = ……

The actual length of the knitting needle, x = ……

Observed distance between the mirror and object needle when the knitting needle is placed between them, y = …….

Observed distance between the mirror and image needle when the knitting needle is placed between them, z = ……..

Index correction for the object distance, u = x – y = ±…….

Index correction for the image distance, v = x -z = ±…….

Table for u, v, 1/u, and 1/v

Distance u (cm) 12 16 20
Distance v (cm)      
     

Calculations

Calculating the Focal Length Using the Graphical Method:

A u – v graph is plotted with u on the x-axis and v on the y-axis. According to the sign conventions, both u and v are negative. The points for different values of u and v are plotted using the observation table. The resulting graph is a rectangular parabola.

A line OA is drawn on either axis at an angle of 45°, intersecting at point A. Perpendiculars AB and AD are drawn on the x and y axes. At point A, the values of u and v are the same. The coordinates at A are (2f, 2f) as u and v are the same for a concave mirror when the object is placed at the centre of curvature. Hence, u = v = R = 2f.

Explanation

  • Applying the mirror formula to point A, we get which can be rewritten as u = v,

Therefore, the focal length of the concave mirror is obtained from the coordinates of A.

∴ Mean value of f = -…….

  • Both these values are negative according to the sign convention. The resulting graph is a straight line and is plotted in the third quadrant. Both axes are intersected by a straight line at 45°, making equal intercepts. The focal length is determined by measuring the distance of OA and OB,

  • For the u-v graph, select a suitable scale to represent u along the X’-axis and v along the Y’-axis. Mark distances such as u1, u2,…. along the OX’-axis and corresponding distances such as v1, v2,….. along the OY’-axis. Draw straight lines to join u1 with v1 and other corresponding points. All these straight lines intersect at point K. Draw perpendiculars KL and KM on the X’ and Y’ axes.

Then -f = OL = OM

∴ f = -…… cm

Findings

The focal length of the given concave mirror is determined from the following:

  • Focal length is calculated as, = ……..cm
  • (u – v) graph = -……cm
  • graph = -………cm
  • (u – v) graph = -…….cm

Precautionary Measures
  • The uprights should be vertical.
  • The principal axis of the mirror should be perpendicular, and the central line of the optical bench should be parallel.
  • Ensure your eye is at least 30 cm away from the needle when locating the position of the image.
  • Remove the parallax tip-to-tip between the needle and the image obtained from the needle.
  • The tips of the object and image should be at the same height relative to the concave mirror's pole.
  • Apply index correction for u and v.

Potential Errors
  • Parallax removal may not be perfect.
  • The uprights may not be vertical.

Viva Questions

Q1. What is the lens formula?

Answer: The lens formula is an equation that explains the relationship between the object distance (u), the image distance (v), and the lens's focal length (f).

Q2. What is the optical center of a lens?

Answer: The optical center of a lens is a fixed point inside the lens through which a light ray passes along the principal axis.

Q3. What is the principal axis of a lens?

Answer: The principal axis of a lens is a straight line that passes through the lens's center of curvature.

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Frequently Asked Questions

Lens formula is defined as the equation which explains the relation between object distance (u), image distance (v), and focal length of the lens (f).

Optical centre of a lens is defined as the fixed point inside the lens on the principal axis through which light ray passes.

Principal axis of a lens is the straight line passing through the centre of curvature of the lens.
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