Determining the focal length of a convex mirror, which always produces virtual images, can be challenging directly. However, by strategically using a convex lens, we can create a real image that acts as a virtual object for the convex mirror, enabling the precise determination of its focal length. This method leverages the principle of light rays re-tracing their path when incident normally on a spherical mirror.
Understanding the Principle
The core idea is to make light rays converge towards the center of curvature of the convex mirror. When light rays strike a spherical mirror along its normal (i.e., passing through its center of curvature), they reflect back along the same path. A convex lens is used to create a real image that serves as this converging point for the convex mirror.
The general steps involve:
- Forming a Real Image: A convex lens is used to form a real, inverted image of a distant or nearby object. This image will serve as the virtual object for the convex mirror.
- Auto-Collimation: The convex mirror is then placed in the path of these converging rays. By carefully adjusting the mirror's position, we can ensure that the rays, after reflection from the mirror and passing back through the lens, converge to form a final image that coincides with the original object. This "image formed in the same location" (as observed in practical demonstrations using a candle) indicates that the rays struck the convex mirror normally.
- Measuring Radius of Curvature: When the final image coincides with the object, it implies that the real image formed by the convex lens (which acts as the virtual object for the mirror) is precisely at the convex mirror's center of curvature (C). The distance from the mirror's pole to this point is its radius of curvature (R).
- Calculating Focal Length: The focal length (f) of a spherical mirror is half its radius of curvature (R), so f = R/2. For a convex mirror, the focal length is conventionally considered negative, indicating its virtual focus.
Materials Required
To perform this experiment, you will typically need:
- Optical Bench: A sturdy platform to align the optical components accurately.
- Convex Lens: A lens with a known or easily determinable focal length (e.g., 10-20 cm).
- Convex Mirror: The mirror whose focal length is to be determined.
- Object Pin/Candle: A brightly lit object, such as an illuminated pin or a candle flame (as used in practical setups), to act as the source.
- Screen/Cross-Wire: For accurately locating image positions.
- Meter Scale: For precise measurements of distances.
- Lens and Mirror Holders: To secure the components on the optical bench.
Step-by-Step Procedure
Follow these steps to determine the focal length of a convex mirror using a convex lens:
Part 1: Forming the Intermediate Real Image
- Preliminary Setup: Mount the object pin (or candle) at one end of the optical bench and the convex lens in a holder. Ensure the lens is perpendicular to the optical axis.
- Forming Image I₁: Adjust the position of the convex lens until a sharp, inverted, real image (let's call it I₁) of the object pin is formed on a screen placed on the optical bench. Note the exact positions of the object (O), the convex lens (L), and the image I₁ on the optical bench. This image I₁ will serve as the virtual object for the convex mirror.
- Practical Tip: To avoid parallax error, use a cross-wire or a semi-transparent screen to precisely locate I₁.
Part 2: Incorporating the Convex Mirror for Auto-Collimation
- Introducing the Convex Mirror: Without disturbing the positions of the object (O) and the convex lens (L), place the convex mirror (M) between the convex lens (L) and the position where I₁ was formed. Ensure the reflecting surface of the convex mirror faces the convex lens.
- Achieving Coincidence: Slowly adjust the position of the convex mirror (M) along the optical bench. Observe the light rays passing through the lens, reflecting off the convex mirror, and then passing back through the lens. Continue adjusting the mirror until a final, sharp image (I₂) is formed that coincides exactly with the original object (O).
- Reference Insight: This is the "image formed in the same location" phenomenon observed in practical setups, such as when using a candle as the object, where the final image appears at the candle itself.
- Crucial Condition: When I₂ coincides with O, it signifies that the light rays, after striking the convex mirror, are diverging as if they originated from the original object. This specific condition implies that the real image I₁ (formed by the convex lens alone) was acting as a virtual object located precisely at the center of curvature (C) of the convex mirror.
Part 3: Measurements and Calculation
- Measure Distances:
- Note the exact position of the convex mirror (M) on the optical bench when the final image (I₂) coincides with the object (O).
- Recall the noted position of the intermediate image (I₁) from Part 1.
- The distance between the position of the convex mirror (M) and the position of the intermediate image (I₁) is the radius of curvature (R) of the convex mirror. Let this be
MC.R = |Position of I₁ - Position of M|
- Calculate Focal Length: The focal length
fof the convex mirror is half its radius of curvature.f = R / 2- Since it's a convex mirror, its focal length is conventionally negative. Therefore, the focal length of the convex mirror is
-R / 2.
Table of Observations (Example)
| S. No. | Position of Object (O) (cm) | Position of Convex Lens (L) (cm) | Position of Intermediate Image (I₁) (cm) | Position of Convex Mirror (M) (cm) | Radius of Curvature, R = |I₁ - M| (cm) | Focal Length, f = -R/2 (cm) |
| :----: | :-------------------------: | :------------------------------: | :-------------------------------------: | :---------------------------------: | :---------------------------------: | :--------------------------: |
| 1 | 0.0 | 15.0 | 35.0 | 25.0 | 10.0 | -5.0 |
| 2 | 0.0 | 18.0 | 40.0 | 30.0 | 10.0 | -5.0 |
| 3 | 0.0 | 12.0 | 30.0 | 20.0 | 10.0 | -5.0 |
| | | | | | Mean R: X.X (cm) | Mean f: -X.X (cm) |
Note: The values in the table are illustrative. Actual experimental values will vary.
Practical Considerations and Accuracy Tips
- Proper Alignment: Ensure all components (object, lens, mirror) are perfectly centered and aligned on the optical bench. Misalignment can lead to distorted images and inaccurate readings.
- Sharp Image: Always aim for the sharpest possible image by fine-tuning the positions. Use the parallax method to confirm the image position.
- Multiple Readings: Take several sets of readings by slightly varying the initial position of the convex lens or object. This helps in minimizing random errors and obtaining a more reliable average value for the focal length.
- Well-Lit Object: A bright object (like an incandescent bulb or an LED source with a cross-wire) yields a clearer image, making it easier to locate.
- Avoid External Light: Perform the experiment in a dimly lit room to enhance image visibility.
By meticulously following these steps, you can accurately determine the focal length of a convex mirror using a convex lens, a fundamental experiment in optics.
