Concave Mirror Experiment
1. Aim
To find the value of image distance (v) for different values of object distance (u) in the case of a concave mirror and to find the focal length of the mirror.
2. Apparatus Used
- Concave mirror with holder
- Optical bench with scale
- Two optical riders
- Object pin or needle (illuminated object)
- Screen or white card
- Half meter scale
- Clamps and stands
- Plumb line
3. Diagram
4. Theory
A concave mirror is a spherical mirror whose reflecting surface is curved inward, i.e., away from the incident light. The reflection of light rays from a concave mirror follows these rules:
- A ray passing through the center of curvature gets reflected back along the same path.
- A ray parallel to the principal axis passes through the focus after reflection.
- A ray passing through the focus becomes parallel to the principal axis after reflection.
- A ray incident at the pole gets reflected making the same angle with the principal axis.
The relationship between object distance (u), image distance (v), and focal length (f) for a spherical mirror is given by the mirror equation:
\[\frac{1}{f} = \frac{1}{v} + \frac{1}{u}\]
Or, it can be rearranged as:
\[f = \frac{uv}{u+v}\]
Where:
- f = focal length of the concave mirror
- u = object distance (distance of the object from the pole of the mirror)
- v = image distance (distance of the image from the pole of the mirror)
According to the sign convention:
- All distances are measured from the pole of the mirror.
- Distances measured in the direction of incident light are positive.
- Distances measured against the direction of incident light are negative.
- For a concave mirror, the focal length (f) is negative.
5. Formula
Mirror Equation: \[\frac{1}{f} = \frac{1}{v} + \frac{1}{u}\]
Rearranged for focal length: \[f = \frac{uv}{u+v}\]
For multiple readings, the average focal length: \[f_{avg} = \frac{f_1 + f_2 + f_3 + ... + f_n}{n}\]
6. Procedure
- Place the concave mirror on one end of the optical bench using a mirror holder.
- Place the object pin (illuminated needle) on the other end of the optical bench.
- Place the screen on the bench between the mirror and the object.
- Adjust the position of the screen until a clear, sharp image of the object is formed on it.
- Measure and record the distance of the object from the mirror (u) and the distance of the image from the mirror (v).
- Change the position of the object and repeat steps 4 and 5 for at least five different values of u.
- For each set of u and v, calculate the focal length using the formula \(f = \frac{uv}{u+v}\).
- Find the average value of the focal length.
- Plot a graph between 1/u and 1/v. The slope of this graph gives the value of focal length.
7. Observation Table
| S.No. | Object Distance (u) cm | Image Distance (v) cm | 1/u (cm-1) | 1/v (cm-1) | Focal Length (f = uv/(u+v)) cm |
|---|---|---|---|---|---|
| 1 | |||||
| 2 | |||||
| 3 | |||||
| 4 | |||||
| 5 |
8. Calculations
For each set of readings, calculate:
-
The reciprocal of object distance (1/u):
\[\frac{1}{u} = \frac{1}{object\ distance\ in\ cm}\]
-
The reciprocal of image distance (1/v):
\[\frac{1}{v} = \frac{1}{image\ distance\ in\ cm}\]
-
The focal length for each reading:
\[f = \frac{u \times v}{u + v}\]
-
Calculate the average focal length:
\[f_{avg} = \frac{f_1 + f_2 + f_3 + f_4 + f_5}{5}\]
Additionally, plot a graph between 1/u (on x-axis) and 1/v (on y-axis). This graph should be a straight line. The slope of this line gives the value of focal length.
9. Result
The focal length of the given concave mirror as determined by:
- Direct calculation method: f = ______ cm
- Graphical method: f = ______ cm
The percentage error in the focal length measurement is: ______%
10. Precautions
- The optical bench should be placed on a horizontal and leveled surface.
- The concave mirror should be fixed perpendicular to the optical bench.
- The center of the object pin, mirror, and screen should be at the same height from the bench.
- The object should be well-illuminated to get a clear image.
- The image obtained should be sharp and clear before taking readings.
- Parallax error should be avoided while taking measurements.
- All distances should be measured from the pole of the mirror.
- The object should not be placed at the focus of the mirror as the image will be formed at infinity.
- To ensure accuracy, take multiple readings and find the average value.
- Handle the apparatus with care to avoid any damage.
11. Sources of Error
- Spherical aberration of the mirror may affect the sharpness of the image.
- Parallax error while measuring the distances.
- The object and image may not be exactly on the principal axis.
- The mirror's surface might not be perfectly concave.
- Difficulty in locating the exact position where the sharpest image is formed.
- Error in judging the position of the pole of the mirror.
- Human errors in measurement and recording of data.
- Imperfect alignment of the optical components.
- Environmental factors like poor lighting can affect the clarity of the image.
12. Viva Voice Questions
What is a concave mirror?
A concave mirror is a spherical mirror whose reflecting surface is curved inward (away from the incident light). It converges parallel rays of light to a focus after reflection.
What is the difference between real and virtual images?
A real image is formed when light rays actually converge at a point after reflection or refraction. It can be captured on a screen. A virtual image is formed when light rays appear to diverge from a point after reflection or refraction. It cannot be captured on a screen and can only be seen by looking into the mirror or lens.
Explain the mirror equation and its derivation.
The mirror equation relates object distance (u), image distance (v), and focal length (f) of a spherical mirror: \(\frac{1}{f} = \frac{1}{v} + \frac{1}{u}\)
This equation can be derived using similar triangles formed by the object, image, and the mirror. It follows from the laws of reflection and the geometry of the spherical mirror.
What is the significance of the sign convention in mirror problems?
The sign convention provides a consistent way to handle various scenarios in mirror problems. By following the convention, we can determine the nature (real/virtual) and position of the image. In the Cartesian sign convention:
- All distances are measured from the pole of the mirror
- Distances measured in the direction of incident light are positive
- Distances measured against the direction of incident light are negative
- Heights measured upwards from the principal axis are positive
- Heights measured downwards from the principal axis are negative
Why is the image inverted when the object is placed beyond the center of curvature of a concave mirror?
When an object is placed beyond the center of curvature of a concave mirror, the rays of light after reflection converge below the principal axis (if the object is above the axis). This causes the image to be inverted. Geometrically, the ray diagram shows that the image forms on the opposite side of the principal axis relative to the object, resulting in an inverted image.
What is the relationship between focal length and radius of curvature for a spherical mirror?
For a spherical mirror, the focal length (f) is half of the radius of curvature (R):
\[f = \frac{R}{2}\]
This relationship holds true for both concave and convex mirrors.
Why do we get different values of 'v' for different values of 'u'?
The image distance (v) varies with object distance (u) according to the mirror equation \(\frac{1}{f} = \frac{1}{v} + \frac{1}{u}\). As the object distance changes, the angle at which rays strike the mirror changes, resulting in different paths after reflection and consequently different image positions.
What happens when the object is placed at the focus of a concave mirror?
When an object is placed at the focus of a concave mirror, the reflected rays become parallel to the principal axis after reflection. Theoretically, the image forms at infinity. In practical terms, no real image is formed on a screen when the object is at the focus.
Why is a graph between 1/u and 1/v useful in determining the focal length?
From the mirror equation \(\frac{1}{f} = \frac{1}{v} + \frac{1}{u}\), we can rearrange to get \(\frac{1}{v} = \frac{1}{f} - \frac{1}{u}\). This is in the form of a straight line equation y = mx + c, where y = 1/v, x = 1/u, m = -1, and c = 1/f. When plotted, the y-intercept gives 1/f, from which we can determine the focal length. This graphical method reduces random errors in measurement.
What are the applications of concave mirrors?
Concave mirrors have many applications including:
- Shaving and makeup mirrors (magnification)
- Headlights in vehicles (focusing light)
- Solar furnaces (concentrating solar energy)
- Astronomical telescopes (collecting light)
- Dentist's mirrors (magnification)
- Searchlights (producing parallel beam of light)
- Ophthalmoscopes (examining the eye)