Spherical Mirrors

A spherical mirror is a curved reflective surface that forms part of a sphere and is widely used in optics, physics, and everyday life—such as in car headlights, rear-view mirrors, and shaving mirrors. These mirrors are of two main types: concave and convex. A concave mirror focuses light to a point, making it useful for magnification and precision tasks, while a convex mirror spreads light outward, offering a wider field of view ideal for safety and surveillance. Found in telescopes, microscopes, and solar devices, spherical mirrors are essential in science and technology. This guide covers their structure, principles, and applications, giving you a clear and practical understanding of how they work.

1.0Definition of Spherical Mirror

A spherical mirror is a mirror that has the shape of a piece cut out of a spherical surface. A curved mirror is part of a hollow sphere. If reflection takes place from the inner surface, then the mirror is called concave and if its outer surface acts as reflector it is convex.

2.0Basic Terms Related to Spherical Mirrors

C = Centre of curvature

R = Radius of curvature

P = Pole (geometric centre of mirror)

1.A pole is any point on the reflecting surface of the mirror. For convenience we take it as the central point, P of the mirror.

2.Principal–section is any section of the mirror such as MM' passing through the pole.

3.Centre of curvature is the centre C of the sphere of which the mirror is a part.

4.Radius of curvature is the radius R of the sphere of which the mirror is a part.

5.Principal–axis is the line CP, joining the pole and centre of curvature of the mirror.

Aperture

• It is the effective diameter of the light reflecting area of the mirror.

How to draw the normal in spherical mirror:

• We can draw normal to a spherical mirror at a particular point by joining the particular point with the centre of curvature of the mirror.

Paraxial Rays: Those rays which make a very small angle with normal at point of incidence are called paraxial rays.

Marginal Rays: Those rays which make a large angle with normal at a point of incidence are called marginal rays.

3.0Focal Length of A Spherical Mirror

• When a parallel beam of paraxial rays strikes a concave mirror, the reflected rays converge at the principal focus F. In a convex mirror, they appear to diverge from F on the principal axis.”

Derivation:

$\Delta CNF\sim \Delta NFM$

then, $CN=NM=\frac{R}{2}$

$\cos i=\frac{CN}{CF}=\frac{\left(\frac{R}{2}\right)}{CF}$

$CF=\frac{R}{2\cos i}$

$FP=CP-PF=R-\frac{R}{2\cos i}$

$\text{If }i\text{ is very small i.e. rays are paraxial, then }\cos i\approx 1$

$FP=R-\frac{R}{2}\Rightarrow FP=\frac{R}{2}=f\text{ (Here, }f\text{ is called focal length of mirror)}$

Conclusion:

For Marginal Rays

$FP=R-\frac{R}{2\cos i}$

If rays are paraxial then,

$FP=f=\frac{R}{2}$

When paraxial light rays parallel to the principal axis are incident on a mirror then the point where they meet (concave mirror) or appear to meet (convex mirror) after reflection is known as focus (F).

Focal Plane

• If the parallel paraxial beam of light were incident, making some angle with the principal axis, the reflected rays would converge (or appear to diverge) from a point in a plane through F normal to the principal axis. This is called the focal plane of the mirror.
• If the parallel paraxial beam of light were incident, making some angle with plane passing through focus and perpendicular to the principal axis,

$\tan \alpha =\frac{h}{FP}$

Rays are paraxial so,

$\tan \alpha \approx \alpha$

$\alpha =\frac{h}{f}$

$h=f\alpha$

4.0Rules For Image Formation

(for paraxial rays only, based on the laws of reflection)

1.A light ray parallel to the principal axis after reflection from the mirror passes or appears to pass through its focus (by definition of focus).

2.A light ray passing through or directed towards focus, becomes parallel to the principal axis after reflection from the mirror.

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

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

Image Formation by Concave Mirror

S.No.	Position of Object	Position of Image	Nature	Size	Ray Diagram
1.	Place at infinity	Formed at F	Real, Inverted	Highly diminished
2.	Placed in between infinity and C	Formed in between C and F	Real, Inverted	Diminished
3.	Placed at C	Formed at C	Real, Inverted	Equal in size
4.	Placed in between F and C	Formed beyond C	Real, Inverted	Enlarged
5.	Placed between F and C, very near to F	Formed at negative infinity	Real, Inverted	Highly enlarged
6.	Placed between F and pole, very near to F	Formed at positive infinity	Virtual, erect	Highly enlarged
7.	Placed between F and P	Formed behind the mirror	Virtual, erect	Enlarged

Things to Keep in mind while solving the problems

Note: Concave mirrors always form real images for virtual objects.

Image formation by convex mirror

• Image is always virtual and erect, whatever be the position of the real object.

Keep in mind during solving the problems

•  Convex mirrors form both real & virtual Image of virtual object, depending on the position of virtual object.

5.0Sign Convention & Mirror Formula

The Cartesian sign convention, measuring distances from the pole or optical centre, is used to derive formulas for spherical mirrors and lenses.

• Along the principal axis, distances are measured from the pole (Pole is taken as the origin).
• Distances in the direction of incident light are taken positive while those along

opposite directions are taken negative.

• The distances above the principal axis are taken positive while below it is negative.
• Whenever and wherever possible, incident light is taken to travel from left to right.    

Things to Keep in mind while solving a problem:

Mirror Formula: Relation between u, v and f in a spherical mirror.

An object is placed at a distance u from the pole of a mirror for small angles and its image is formed at a distance v (from the pole).

If angle is very small

$\alpha =\frac{MP}{u},\beta =\frac{MP}{R},\gamma =\frac{MP}{v}$

From $\Delta CMO,\beta =\alpha +\theta \Rightarrow \theta =\beta -\alpha$

From  $\Delta CMI,\gamma =\beta +\theta \Rightarrow \theta =\gamma -\beta$

So we can write $\beta -\alpha =\gamma -\beta \Rightarrow 2\beta =\gamma +\alpha$

$\therefore \frac{2}{R}=\frac{1}{v}+\frac{1}{u}\Rightarrow \frac{1}{f}=\frac{1}{u}+\frac{1}{v}$ This equation is called Mirror Formula

6.0Transverse or Lateral Magnification

• If a one dimensional object is placed perpendicular to the principal axis then ratio of image height and object height is called transverse or lateral magnification.

$m_t=\frac{h_i}{h_o}$

Derivation of Transverse Magnification:

$\frac{h_i}{h_o}=\frac{v}{u}$

Put the value with sign

$u\to -u,v\to -v,h_i\to -h_i,and{h}_o\to +h_o:then$

$-\frac{h_i}{h_o}and=\frac{-v}{-u}$

$\frac{h_i}{h_o}and=-\frac{v}{u}$

$m_t=\frac{h_i}{h_o}and=-\frac{v}{u}$

$\begin{array}{rl}\tan \theta =\frac{h_o}{u}& \text{ and }\tan \theta =\frac{h_i}{v}\\ & \frac{h_i}{h_o}=\frac{v}{u}\end{array}$

7.0Longitudinal Magnification

• If an object is placed along the principal axis then the ratio of length of image and length of object is called longitudinal or axial magnification.

$m_L=\frac{\text{Length of image}}{\text{Length of Object}}=\frac{I_i}{I_0}$

$m_L=\frac{\text{Length of image}}{\text{Length of Object}}=\left|\frac{v_1-v_2}{u_2-u_1}\right|$

Longitudinal magnification for small objects ($L_o\ll f$):

$\text{object length}=du;\text{Image length}=dv$

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

Differentiating w.r.t u

$-\frac{1}{v^2}\frac{dv}{du}-\frac{1}{u^2}=0\Rightarrow \frac{dv}{du}=-\frac{v^2}{u^2}$

If we use only magnitude then $\frac{dv}{du}=\frac{v^2}{u^2}$

$M_L=\frac{I_i}{I_o}=\frac{v^2}{u^2}=m_t^2$

8.0Superficial Magnification

• If a two-dimensional object is placed with its plane perpendicular to the principal axis then its magnification is known as superficial magnification.

$m_s=\frac{A_i}{A_o}$

$A_o=w_o{h}_o\text{ and }{A}_i=w_i{h}_i$

$m_t=\frac{h_i}{h_o}=\frac{w_i}{w_o}$

$\frac{A_i}{A_o}=\frac{w_i{h}_i}{w_o{h}_o}$

$m_t=\frac{h_i}{h_o}=\frac{w_i}{w_o}$

$\frac{A_i}{A_o}=m_t^2=m_s$

9.0Newton’s Formula

• This equation relates object and image distances to the focal length of a mirror or lens.

$x_1$: Object distance from the focus

$x_2$: Image distance from the focus

The relation between $x_1,x_2$, f is 

$f=\sqrt{x_1{x}_2}$

10.0Velocity of Image in Spherical Mirror

Case (1): When an object is moving along the principal axis of a spherical mirror.                 

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

Differentiate with respect to time,

$-\frac{1}{u^2}\frac{du}{dt}+-\frac{1}{v^2}\frac{dv}{dt}=0$

$\frac{1}{v^2}\frac{dv}{dt}=-\frac{1}{u^2}\frac{du}{dt}$

$\frac{dv}{dt}=-\frac{v^2}{u^2}\frac{du}{dt}$

${\vec{V}}_{IM}=-m_t^2{\vec{V}}_{OM}$

All velocities are instantaneous.

Case (2): When the object is moving perpendicular to the principal axis of the spherical mirror.

From equation $\frac{h_i}{h_o}=-\frac{v}{u}$

Differentiate with respect to time,

$\frac{d{h}_i}{dt}=-\frac{v}{u}\frac{d{h}_o}{dt}$

${\vec{V}}_{IM}=m_t{\vec{V}}_{OM}$

All velocities are instantaneous.

Combination of Mirrors

Note: In case of successive reflection from mirrors, the overall lateral magnification is given by $m_1\times m_2\times m_3\dots \dots \dots ,\text{ where }{m}_1,m_2$ etc. are lateral magnifications produced by individual mirrors.

Cutting of Mirror

All the parts of the mirror have the same hollow sphere, so its centre of curvature is the same therefore no of images found is 1. The intensity of the image decreases.