If the object distance and the image distance from a concave mirror is -20 cm, what is the focal length of the mirror?
-10 cm
Calculating Focal Length of a Concave Mirror using the Mirror Formula
The question asks us to find the focal length of a concave mirror given the object distance and the image distance. We are given that the object distance and the image distance from the concave mirror are both -20 cm.
In optics, the sign conventions are very important. For a concave mirror, the object distance (u) is usually taken as negative if the object is placed in front of the mirror (which is standard for real objects). The image distance (v) is taken as negative for a real image formed in front of the mirror and positive for a virtual image formed behind the mirror. The focal length (f) of a concave mirror is always taken as negative.
Given Information:
- Object distance, \(u = -20\) cm
- Image distance, \(v = -20\) cm
Both distances are negative, indicating that the object is real (in front of the mirror) and the image is real (formed in front of the mirror). This scenario typically occurs when the object is placed at the center of curvature (C) of a concave mirror. When an object is placed at the center of curvature, a real, inverted, and same-sized image is formed at the center of curvature.
Applying the Mirror Formula
To find the focal length (\(f\)) of the concave mirror, we use the mirror formula, which relates the focal length, object distance, and image distance:
The mirror formula is:
\(\frac{1}{f} = \frac{1}{v} + \frac{1}{u}\)
Now, we substitute the given values of \(u\) and \(v\) into the formula:
\(\frac{1}{f} = \frac{1}{(-20)} + \frac{1}{(-20)}\)
\(\frac{1}{f} = \frac{-1}{20} + \frac{-1}{20}\)
To add these fractions, they already have a common denominator (20):
\(\frac{1}{f} = \frac{-1 + (-1)}{20}\)
\(\frac{1}{f} = \frac{-2}{20}\)
Simplify the fraction:
\(\frac{1}{f} = \frac{-1}{10}\)
To find \(f\), we take the reciprocal of both sides of the equation:
\(f = -10\) cm
The calculated focal length is -10 cm. The negative sign confirms that it is indeed a concave mirror.
Summary of Calculation Steps:
- Identify the given object distance (u) and image distance (v) with correct signs based on convention.
- Write down the mirror formula: \(\frac{1}{f} = \frac{1}{v} + \frac{1}{u}\).
- Substitute the given values of u and v into the formula.
- Solve the equation for f.
- Interpret the sign of the focal length.
Conclusion on Focal Length
Based on the calculation using the mirror formula and the given object and image distances of -20 cm, the focal length of the concave mirror is -10 cm.
| Variable | Description | Sign Convention (Example) |
|---|---|---|
| \(u\) | Object Distance | Negative (real object in front) |
| \(v\) | Image Distance | Negative (real image in front) Positive (virtual image behind) |
| \(f\) | Focal Length | Negative (for a concave mirror) |
| \(R\) | Radius of Curvature | Negative (for a concave mirror) |
Revision Table: Concave Mirror Formulas
| Formula | Description |
|---|---|
| \(\frac{1}{f} = \frac{1}{v} + \frac{1}{u}\) | Mirror Formula |
| \(R = 2f\) | Relation between Radius of Curvature and Focal Length |
| \(m = \frac{h_i}{h_o} = -\frac{v}{u}\) | Magnification Formula (\(h_i\): image height, \(h_o\): object height) |
Additional Information: Properties of Images formed by a Concave Mirror
The nature and position of the image formed by a concave mirror depend on the position of the object. Here are some key cases:
- Object at Infinity: Image is real, inverted, and highly diminished, formed at the focal point (F).
- Object beyond Center of Curvature (C): Image is real, inverted, and diminished, formed between F and C.
- Object at Center of Curvature (C): Image is real, inverted, and same size, formed at C. This matches the scenario in the problem.
- Object between Focal Point (F) and Center of Curvature (C): Image is real, inverted, and magnified, formed beyond C.
- Object at Focal Point (F): Image is real, inverted, and highly magnified (at infinity), formed at infinity.
- Object between Pole (P) and Focal Point (F): Image is virtual, erect, and magnified, formed behind the mirror.
Understanding these cases helps predict the image properties for different object positions relative to the concave mirror's focal point and center of curvature.
Important Questions from Refraction and Reflection
Which of the following is NOT an example of refraction of light?
Water drops shine on a lotus leaf due to:
A convex lens 'A' of focal length $10 \text{ cm}$ and another convex lens 'B' of focal length $20 \text{ cm}$ are kept along the same axis with a distance '$d$' between them. If a parallel beam of light falling on 'A' leaves 'B' as a parallel beam, then the distance '$d$' in $cm$ will be :
- A ray is incident at an angle of incidence $i$ on one surface of a small angle prism (with angle of prism $A$ and refractive index $\mu$). The ray emerges normally from the opposite surface, causing a total angle of deviation $\delta$ from its original path. Assuming all angles are small, the angle of incidence $i$ is nearly equal to:
- A light ray travels from transparent medium A to transparent medium B, separated by a plane boundary.
Medium A has a refractive index of $n_A = 1.6$. The speed of light in medium B is $v_B = 2.4 \times 10^8 \text{ m/s}$.
Given the speed of light in vacuum is $c = 3.0 \times 10^8 \text{ m/s}$, the critical angle for total internal reflection when light passes from medium A to medium B is:
