Convex Mirror

Spherical mirrors are reflective surfaces shaped from a hollow sphere, made of glass or plastic. They are classified into two types: concave mirrors, which curve inward like the inside of a sphere, and convex mirrors, which curve outward, resembling the outer surface.A convex mirror is a spherical mirror characterized by its outward-curving reflective surface. This distinctive design enables convex mirrors to offer a broader field of view, making them suitable for numerous applications. Additionally, convex mirrors are crucial for studying image formation and reflection principles, as they produce virtual images that are upright and smaller than the actual object, which makes them an important topic in physics.

1.0Definition of Convex Mirror

• It consists of a section of a hollow sphere, with the inner surface (depressed side) coated with a reflective material, while the outer surface (bulging side) serves as the reflecting surface.
• In convex mirrors, light rays that are parallel to the principal axis, after reflection, seem to diverge from a common point along the principal axis known as the principal focus (F).
• A convex mirror is referred to as a divergent mirror because it causes light rays to spread apart

2.0Rules of Image Formation(Ray Diagram)

1. A light ray parallel to the principal axis, upon reflecting off the mirror, either passes through or appears to originate from its focus.

2. A light ray that travels through or is directed toward the focus becomes parallel to the principal axis after reflecting off the mirror.

3. A ray passing through or directed towards the centre of curvature, retraces its path (as for it $\angle i=0$ and so $\angle r=0$) after reflection from the mirror.

4. Incident and reflected rays at the pole of a mirror are symmetrical about the principal axis $\angle i=\angle r$

3.0Sign Convention 

• Along the principal axis,distance is measured from the pole.
• Distances measured  in the direction of incident light are taken positive while those along opposite directions are taken negative.
• The distance above the principal axis is taken positive while below is negative.
• Whenever and wherever possible, incident light is taken to travel from left to right.

Note: Things to keep in mind while solving a problem.

4.0Mirror Formula(Convex Mirror)

• AB be an object lying on the principal axis of a convex mirror of small aperture, A’B’ is the virtual image of the object.

$△ABC\text{ and }△A^{\prime }{B}^{\prime }C$ are similar

$\frac{A^{\prime }{B}^{\prime }}{AB}=\frac{C{A}^{\prime }}{CA}$ . . . . . . . . . . . .(1)

$△ABP\text{ and }{A}^{\prime }{B}^{\prime }P$

$\frac{A^{\prime }{B}^{\prime }}{AB}=\frac{P{A}^{\prime }}{PA}$ . . . . . . . . . . . . . . . .(2)

 From equation (1) and (2)

$\frac{C{A}^{\prime }}{CA}=\frac{P{A}^{\prime }}{PA}$

$\frac{PC-P{A}^{\prime }}{PC+P{A}^{\prime }}=\frac{P{A}^{\prime }}{PA}$ . .. . . . . . . (3)

Applying sign conventions, P C=R,$P{A}^{\prime }=v$, P A=-u

From using equation (3),we get

$\frac{R-v^{\prime }}{R-u^{\prime }}=\frac{v^{\prime }}{-u}$

$-uR+uv=vR-vu$

$vR+uR=2vu$

$\frac{1}{v}+\frac{1}{u}=\frac{2}{R}$

$\therefore R=2f$

$\frac{1}{v}+\frac{1}{u}=\frac{1}{f}$

5.0Linear Magnification of Convex Mirror

• It is defined as the magnitude of the size of the image to the magnitude of the object$m=\frac{I}{o}=-\frac{v}{u}.$
• $m=\frac{f}{f-u}$
• $m=\frac{f-v}{f}$

6.0Application and its Uses

1. It is utilized as a reflector for street lighting applications.
2. It is employed in large stores to observe customer activities.
3. It is used as a rearview mirror because it produces upright, diminished images. The smaller size of the image expands the field of view behind the vehicle, which is why it is also referred to as a driver's mirror.

7.0Sample Questions On Convex Mirror

Q-1. Demonstrate that a convex mirror consistently forms a virtual image, regardless of the object's position.

Solution :

$\frac{1}{u}+\frac{1}{v}=\frac{1}{f}$

$\frac{1}{v}=\frac{1}{f}-\frac{1}{u}$

Since f > 0 and u < 0 for concave mirror

v > 0

So the image is on the right side of the convex mirror (virtual) whatever be the position of the object.

Q-2. Find out position, nature and size of image of the object as shown in fig.

Solution:

$u=-15\mathrm{cm},f=15\mathrm{cm}$

$m=\frac{f}{f-u}\Rightarrow m=\frac{15}{15-(-15)}=\frac{1}{2}$

$m=-\frac{v}{u}\Rightarrow \frac{1}{2}=\frac{-v}{-15}$

$v=7.5\mathrm{cm}$

$h_i=m\times h_0$

$h_i=\frac{1}{2}\times 2=1\mathrm{cm}$

Image is virtual erect and smaller than the object.

Q-3. A 5 cm long needle is positioned 10 cm in the head of a convex mirror with a focal length of 40 cm. Identify the position, nature, and size of the image of the needle. Additionally, what occurs to the size of the image when the needle is moved further away from the mirror?

Solution:

$u=-10\mathrm{cm}andf=40\mathrm{cm}$

$\frac{1}{v}+\frac{1}{u}=\frac{1}{f}$

$\frac{1}{v}-\frac{1}{10}=\frac{1}{40}\Rightarrow \frac{1}{v}=\frac{5}{40}=\frac{1}{8}\Rightarrow v=0.8\mathrm{cm}$

$\Rightarrow$ As the needle is moved away, the image shifts behind the mirror towards the focus, and its size continues to decrease. When the needle is positioned far away, it nearly forms a point image at the focus.