Sign Convention for Reflection by Spherical Mirrors
New Cartesian Sign Convention
- In this convention, the pole (P) of the mirror is taken as the origin. The principal axis of the mirror is taken as the x-axis (X’X) of the coordinate system.
The conventions are as follows:
- The object is always placed to the left of the mirror, i.e., light from the object falls on the mirror from the left-hand side.
- All distances parallel to the principal axis are measured from the pole of the mirror.
- All distances measured to the right of the origin (along + x-axis) are taken as positive, while those measured to the left of the origin (along – x-axis) are taken as negative.
- Distances measured perpendicular to and above the principal axis (along + y-axis) are taken as positive.
- Distances measured perpendicular to and below the principal axis (along – y-axis) are taken as negative.
Sign conventions are applied to obtain the mirror formula and solve related numerical problems.
Mirror Formula and Magnification
- In a spherical mirror, the distance of the object from its pole is called the object distance (u).
- The distance of the image from the pole of the mirror is called the image distance (v).
- The distance of the principal focus from the pole is called the focal length (f).
- This formula is valid in all situations for all spherical mirrors for all positions of the object.
Magnification (m)
- It is the enlargement of the image formed by a spherical mirror, relative to the size of the object.
- It is the ratio of the height of the image (h′) to the height of the object (h).
- Magnification is also related to the object distance (u) and image distance (v). It can be expressed as:
- The height of the object is taken to be positive as the object is placed above the principal axis.
- The height of the image is taken as positive for virtual images and negative for real images.
- A negative sign in the value of the magnification indicates that the image is real. A positive sign indicates that the image is virtual.
Problem 1:
- A convex mirror used for rear-view on an automobile has a radius of curvature of 3.00 m. If a bus is located at 5.00 m from this mirror, find the position, nature, and size of the image.
Solution
- Radius of curvature, R = +3.00 m
- Object-distance, u = –5.00 m
- Image-distance, v = ?
- Height of the image, h′ = ?
- Focal length, f = R/2 = +3.00 m / 2 = +1.50 m (as the principal focus of a convex mirror is behind the mirror)
- The image is 1.15 m at the back of the mirror.
- The image is virtual, erect, and smaller by a factor of 0.23.
Problem 2:
- An object, 4.0 cm in size, is placed at 25.0 cm in front of a concave mirror of focal length 15.0 cm. At what distance from the mirror should a screen be placed in order to obtain a sharp image? Find the nature and the size of the image.
Solution
- Object-size, h = +4.0 cm
- Object-distance, u = –25.0 cm
- Focal length, f = –15.0 cm
- Image-distance, v = ?
- Image-size, h′ = ?
- v = –37.5 cm
- The screen should be placed at 37.5 cm in front of the mirror. The image is real.
- Height of the image, h′ = –6.0 cm
- The image is inverted and enlarged.
Select a Topic 👇
1. Reflection of Light 2. Spherical Mirrors 3. Image Formation by Spherical Mirrors 4. Sign Convention, Mirror Formula & Magnification 5. Refraction of Light 6. Refraction through a Rectangular Glass Slab 7. Refractive Index, Refraction by Spherical Lenses 8. Image Formation by Lenses 9. Lens Formula & Magnification, Power of Lenses