Mirror formula
Understanding the Mirror Formula
The mirror formula is a fundamental concept in the field of optics, specifically in the study of mirrors. It relates the object distance ($u$), the image distance ($v$), and the focal length ($f$) of a spherical mirror. This formula is applicable to both concave and convex mirrors and is essential for solving problems related to image formation.
The Mirror Formula
The mirror formula is given by:
[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} ]
Where:
- $f$ is the focal length of the mirror.
- $v$ is the distance from the mirror to the image.
- $u$ is the distance from the mirror to the object.
Sign Convention
To apply the mirror formula correctly, one must adhere to a specific sign convention. The following rules are generally used:
- All distances are measured from the pole of the mirror.
- The focal length ($f$) is positive for concave mirrors and negative for convex mirrors.
- The object distance ($u$) is always negative since the object is placed to the left of the mirror.
- The image distance ($v$) is positive if the image is formed on the same side as the object (real image) and negative if the image is formed on the opposite side (virtual image).
Differences Between Concave and Convex Mirrors
| Feature | Concave Mirror | Convex Mirror |
|---|---|---|
| Shape | Curved inwards | Curved outwards |
| Focal Length ($f$) | Positive | Negative |
| Real Image | Formed when the object is outside the focal point | Not applicable |
| Virtual Image | Formed when the object is inside the focal point | Always formed |
| Image Orientation | Inverted for real images, Upright for virtual images | Always upright |
| Image Size | Can be larger or smaller than the object | Always smaller than the object |
Examples
Example 1: Image Formation by a Concave Mirror
An object is placed 30 cm in front of a concave mirror with a focal length of 15 cm. Find the image distance and the nature of the image.
Using the mirror formula:
[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} ]
Given $u = -30$ cm (since the object is in front of the mirror) and $f = 15$ cm (positive for concave mirrors), we can find $v$:
[ \frac{1}{15} = \frac{1}{v} - \frac{1}{30} ]
Solving for $v$ gives:
[ \frac{1}{v} = \frac{1}{15} + \frac{1}{30} = \frac{2}{30} + \frac{1}{30} = \frac{3}{30} = \frac{1}{10} ]
Hence, $v = 10$ cm. Since $v$ is positive, the image is real and formed on the same side as the object.
Example 2: Image Formation by a Convex Mirror
An object is placed 20 cm in front of a convex mirror with a focal length of -10 cm. Find the image distance and the nature of the image.
Using the mirror formula:
[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} ]
Given $u = -20$ cm (since the object is in front of the mirror) and $f = -10$ cm (negative for convex mirrors), we can find $v$:
[ \frac{1}{-10} = \frac{1}{v} - \frac{1}{20} ]
Solving for $v$ gives:
[ \frac{1}{v} = \frac{1}{-10} + \frac{1}{20} = -\frac{2}{20} + \frac{1}{20} = -\frac{1}{20} ]
Hence, $v = -20$ cm. Since $v$ is negative, the image is virtual and formed on the opposite side of the object.
Conclusion
The mirror formula is a powerful tool in the study of optics, allowing us to determine the relationship between the object distance, image distance, and focal length of spherical mirrors. By understanding the sign convention and applying the formula correctly, one can predict the properties of the image formed by concave and convex mirrors.