A concave mirror of radius of curvature 50 cm is used to form an image of an object kept at a distance of 25 cm from the mirror on its principal axis. What will be the position of the image from the mirror?
At infinity
Understanding Image Formation with a Concave Mirror
This question asks us to determine the position of the image formed by a concave mirror given its radius of curvature and the object distance. We will use the mirror formula and the relationship between the radius of curvature and the focal length of a spherical mirror.
Given Information
- Type of mirror: Concave mirror
- Radius of curvature (\(R\)): 50 cm
- Object distance from the mirror (\(u\)): 25 cm
Finding the Focal Length
For any spherical mirror, the focal length (\(f\)) is half of its radius of curvature (\(R\)). That is, \( f = \frac{R}{2} \). For a concave mirror, the focal length is considered negative as it is a real focus located in front of the mirror.
So, the magnitude of the focal length is:
\( |f| = \frac{50 \text{ cm}}{2} = 25 \text{ cm} \)
Applying the sign convention for a concave mirror, the focal length is:
\( f = -25 \text{ cm} \)
Applying the Mirror Formula
The mirror formula relates the focal length (\(f\)), the object distance (\(u\)), and the image distance (\(v\)):
\( \frac{1}{f} = \frac{1}{u} + \frac{1}{v} \)
We need to use the correct sign conventions for the object distance as well. The object is placed 25 cm from the mirror, and for a real object in front of the mirror, the object distance (\(u\)) is taken as negative.
So, \( u = -25 \text{ cm} \).
Now, substitute the values of \( f \) and \( u \) into the mirror formula:
\( \frac{1}{-25} = \frac{1}{-25} + \frac{1}{v} \)
Calculating the Image Position
We want to find the image distance \( v \). Let's rearrange the equation to solve for \( \frac{1}{v} \):
\( \frac{1}{v} = \frac{1}{-25} - \frac{1}{-25} \)
\( \frac{1}{v} = \frac{-1}{25} - \left( \frac{-1}{25} \right) \)
\( \frac{1}{v} = \frac{-1}{25} + \frac{1}{25} \)
\( \frac{1}{v} = 0 \)
To find \( v \), we take the reciprocal of both sides:
\( v = \frac{1}{0} \)
This indicates that the image is formed at infinity.
Interpretation of the Result
The object distance (\( u = 25 \text{ cm} \)) is equal to the magnitude of the focal length (\( |f| = 25 \text{ cm} \)). When an object is placed exactly at the focal point of a concave mirror, the rays of light from the object, after reflection from the mirror, become parallel to the principal axis. Parallel rays are considered to meet at infinity, thus forming an image at infinity.
Revision Table: Concave Mirror Key Points
| Concept | Description | Sign Convention (usually) |
|---|---|---|
| Focal Length (f) | Distance from pole to principal focus | Negative for concave mirror |
| Radius of Curvature (R) | Distance from pole to center of curvature | Negative for concave mirror (\(R=2f\)) |
| Object Distance (u) | Distance from pole to object | Negative for real object in front |
| Image Distance (v) | Distance from pole to image | Negative for real image (in front) Positive for virtual image (behind) |
| Mirror Formula | Relates f, u, and v | \( \frac{1}{f} = \frac{1}{u} + \frac{1}{v} \) |
Additional Information: Image Formation at Infinity
The case where the object is placed at the focal point (\( u = f \)) and the image is formed at infinity (\( v = \infty \)) is a specific and important scenario for concave mirrors. This principle is used in devices like searchlights, headlights, and flashlights. A light source (like a bulb) is placed at the focal point of a concave reflector. The light rays diverging from the source hit the reflector and are reflected as a strong, parallel beam of light that travels over a long distance, effectively forming an image at infinity.
Conversely, if the object is at infinity (like distant stars), the image is formed at the focal point. This is why telescopes use concave mirrors to form images of distant objects at their focal plane.
Similar Questions
Which of the following combinations of source and screen would produce the sharpest shadow of an opaque object?
Among the following types of glasses, which one is used for making optical instruments?
In total internal reflection, the light travels from ________.
The energy of a photon, whose momentum is 10 MeV/c, where c is the speed of light, is- given by
In a periscope, the two plane mirrors are kept
The optical phenomenon that is primarily responsible for the observation of rainbow on a rainy day is ________.
Light travels in a straight line (rectilinear propagation of light). This statement does hold if the medium of travel for light is
A myopic person has a power of -1.25 Dioptre. What is the focal length and nature of his lens?
- In a simple astronomical telescope, the objective and the eyepiece used respectively, are:
Which one of the following statements about the aperture of a convex lens is correct?
Important Questions from Optics
Who was the first one to use a glass prism to obtain the spectrum of sunlight?
Twinkling of stars is due to:
In which a convex mirror is used?
A. Rear View mirrors in vehicles
B. Glass windows
C. Makeup Mirror
D. Kaleidoscope
What is the reason for formation of Mirage in desert?
An object is placed 10 cm in front of a lens. The image formed is real, inverted and of same size as the object. What is the focal length and nature of the lens?
