Sign Convention in Spherical Mirrors: An Overview
Sign convention is nothing complicated but a simple way of making calculations simpler and easier by specifying certain rules while dealing with calculations for spherical mirror problems. These rules are a bit different when it comes to dealing with lenses, which we will be discussing later in this article as well. During sign convention application, we should always keep in mind what we are using them for since we already know that these will be different for lenses and spherical mirrors. In sign convention, we assign areas around our lens or mirror, a positive or negative sign and do our further calculations accordingly, keeping in mind the sign allocated to a particular region.
What is a Sign Convention?
The sign convention is allocating positive or negative signs to regions around the mirror or lens, i.e., left or right and up or down of the principal axis of the mirror or lens, in order to make calculations easier and better to understand.
Sign Convention for Mirrors
Sign Convention for Mirror
The distances which are measured along the direction of the incident ray are taken as positive.
The distances which are measured opposite the direction of the incident ray are taken as negative.
The region above the principal axis is considered positive.
The region below the principal axis is considered negative.
All measurements should be taken from the Pole of the mirror.
From the above points, we can conclude about Sign convention for spherical mirrors that:
The focal length of the concave mirror will always be negative since it is on the left of the mirror (in front of the mirror). Similarly, the focal length of the convex mirror will always be positive since it is on the right side of the mirror (behind the mirror).
Object distance will also be negative.
If a real image is formed, then it will form in front of the mirror, hence it will be negative.
If a virtual image is formed, then it will be formed behind the mirror, hence it will be taken as positive.
Since the object is always taken above the principal axis, we will take the object height as positive.
If the image formed is erect, then it will be above the principal axis, therefore its height should be taken as positive.
If the image formed is inverted, then it is formed below the principal axis, hence the image height should be taken as negative.
Sign Convention for Lenses
Sign Convention for Lens
The region on the right of the principal axis of the lens is positive.
The region on the left of the principal axis of the lens is negative.
The region above the principal axis is considered positive.
The region below the principal axis is considered negative.
All measurements should be taken from the Pole of the lens.
From the above convention, we can conclude that:
The focal length of a convex lens is always positive.
The focal length of a concave lens is always taken as negative.
The object taken is always in front of the lens, hence its distance will always be taken as negative.
If a real image is formed, then its distance will be taken as positive since it is being formed in the direction of the incident ray.
If a virtual image is formed, then its distance will be taken as negative since it is being formed opposite to the direction of the incident ray.
Coming to the object will be above the principal axis, hence take its height always as positive.
If the image is erect, then it will be above the principal axis, hence height will be taken as positive.
If the image formed is inverted, then it will be below the principal axis, hence it will be taken as negative.
Some Important Questions
Can a real image be seen in the concave mirror?
Ans: When we place an object further away from the focal point of the mirror, a real image will be formed. But this image formed will be inverted.
What will be the sign convention for concave mirrors?
Ans: The image formed can be real or virtual in the case of the concave mirror. So, the image distance can be positive or negative. So, the focal length can also be positive or negative.
What will be the sign convention for convex lenses?
Ans: Convex mirrors always form virtual and erect images. So, for virtual images, the image distance is positive. And therefore, the focal length is also positive.
Key Features
We use sign convention to make our calculations and understanding easier.
During calculations regarding mirror and lens, we should keep in mind the sign convention used to get correct results.
The focus of the mirror and lens gets changed by changing the sign convention.
Sign convention for focus is different for mirrors and lenses, hence it should be kept in mind whether dealing with a mirror or a lens.
Finally, there are some important questions so as to clear our understanding of the topic.
FAQs on Sign Convention for Mirrors
1. What are the fundamental rules of the Cartesian sign convention used for spherical mirrors as per the CBSE syllabus?
According to the Cartesian sign convention for spherical mirrors, a set of rules is followed to ensure uniformity in calculations. The key rules are:
- All distances are measured from the pole (P) of the mirror, which is treated as the origin (0,0).
- The principal axis of the mirror is considered the x-axis of the coordinate system.
- Distances measured in the same direction as the incident light (usually left to right) are taken as positive.
- Distances measured against the direction of the incident light are taken as negative.
- Heights measured upwards and perpendicular to the principal axis are positive.
- Heights measured downwards and perpendicular to the principal axis are negative.
2. Why is the object distance (u) always considered negative in problems involving spherical mirrors?
The object distance (u) is always taken as negative because, by convention, the object is always placed to the left of the mirror. The incident light travels from the object towards the mirror's pole (from left to right). Since the distance from the pole to the object is measured against the direction of the incident light, its sign is negative.
3. How do you determine the sign of the focal length (f) for concave and convex mirrors?
The sign of the focal length (f) depends on the type of spherical mirror:
- For a concave mirror, the principal focus lies in front of the mirror (on the left side). Since this distance is measured against the direction of incident light, its focal length is always negative.
- For a convex mirror, the principal focus is located behind the mirror (on the right side). This distance is measured in the same direction as the incident light, so its focal length is always positive.
4. What do the positive and negative signs of the image distance (v) tell us about the image's nature?
The sign of the image distance (v) is crucial for understanding the nature and location of the image formed:
- A negative image distance (-v) indicates that the image is formed in front of the mirror. Such images are real and inverted.
- A positive image distance (+v) indicates that the image is formed behind the mirror. These images are virtual and erect.
5. What is the importance of the sign (positive or negative) in the formula for magnification (m)?
The sign of magnification (m) for a spherical mirror directly describes the orientation of the image relative to the object. A positive magnification (+m) means the image is virtual and erect (upright). A negative magnification (-m) means the image is real and inverted (upside down). The value of 'm' indicates the size; if |m| > 1, the image is enlarged, and if |m| < 1, it is diminished.
6. How does the application of sign convention differ between a concave mirror and a convex mirror for an object placed in front of them?
While the object distance (u) is always negative for both, the key differences in sign convention arise for focal length and image distance. For a concave mirror, the focal length (f) is negative. The image distance (v) can be either negative (for real images) or positive (for a virtual image). In contrast, for a convex mirror, the focal length (f) is always positive, and it consistently forms a virtual image, so the image distance (v) is also always positive.
7. If an object is placed at the centre of curvature (C) of a concave mirror, how is the sign convention applied to determine the image properties?
When an object is at the centre of curvature (C) of a concave mirror, we apply the sign convention as follows:
- The object distance is the radius of curvature, so u = -R (negative as it's in front).
- We also know that for a concave mirror, f = R/2, so f = -R/2.
- Using the mirror formula 1/v + 1/u = 1/f, we get 1/v + 1/(-R) = 1/(-R/2).
- This simplifies to 1/v = -2/R + 1/R = -1/R. Therefore, the image distance v = -R.
- The negative sign for 'v' indicates the image is real and inverted, and it is formed at the same position as the object (the centre of curvature).
