Ray Optics Formula Sheet for Class 12: Essential Equations & Concepts

Ray Optics class 12 Formula Sheet

Below is the complete formula list of ray optics is given.

Law of Reflection

• Incident ray, reflected ray and normal to reflecting surface at the point of incidence lie in the same plane.
• Angle of incidence is equal to angle of reflection.

Sign-convention

• In sign convention, all distances measured in the same direction as the incidence ray are taken positive and those measured in the direction opposite of the incident ray are taken negative.
• The heights taken above the principal axis are positive and below negative.

Focal Length of Spherical Mirrors

• The distance between focus and pole of a mirror is called focal length.
• Focal length is equal to half of radius of curvature of the curved spherical mirror.

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

Image Formation by Spherical Mirrors

• The image by a mirror is real if rays after reflection actually meet and virtual if rays are not actually meeting but appear to diverge from a point.
• An incident ray passing through the centre of curvature of mirror retraces its path.
• Mirror Equation: $\frac{1}{u}+\frac{1}{v}=\frac{1}{f}$
• Magnification formula: $\begin{aligned}
  m & =-\frac{v}{u}=\frac{f}{f-u} \\
  & =\frac{f-v}{f}
  \end{aligned}$

Refraction of Light

• When a beam of light encounters another transparent medium, part of light is reflected back. This called internal reflection. The rest of light enter other medium.
• When light is incident obliquely, its propagation direction changes in other medium, this phenomenon is called refraction.
• Red light travels faster than blue light in same medium.

Law of Refraction

• The incident ray, refracted ray and normal to interface at the point of incidence,all lie in same plane.
• The ratio of sine of angle of incidence to the sine of angle of refraction is constant.
• Law of refraction is also called snell's law:

Total Internal Reflection

• If angle of incidence, for light traveling from denser to rarer medium is greater than certain angle called critical angle for the media, no light is transmitted.

$\sin i_c=\frac{1}{n_{12}}$

Here: $n_{12} $ is refractive index of denser medium w.r.t rarer medium.

Refraction at Spherical Surfaces

When light passes through a spherical surface separating two media of different refractive indices, it bends according to Snell’s law.

General Formula for Refraction at Spherical Surfaces

$\frac{n_2}{v}-\frac{n_1}{u}=\frac{n_2-n_1}{R}$

Where:
- $n_1$ : Refractive index of the medium where the object is placed.
- $n_2$ : Refractive index of the second medium.
- $u$ : Object distance from the spherical surface.
- $v$ : Image distance from the spherical surface.
- R: Radius of curvature of the spherical surface (positive if the center of curvature is on the side of the outgoing light).

Special Cases

Case 1: Plane Surface
When the surface is plane $(R=\infty)$ :

$
\frac{n_2}{v}-\frac{n_1}{u}=0 \quad \Longrightarrow \quad v=\frac{n_2}{n_1} u
$

This indicates that the image distance is directly proportional to the object distance.
Case 2: Refractive Index of One Medium is Air
If $n_1=1$ (air) and $n_2=n$ :

$
\frac{n}{v}-\frac{1}{u}=\frac{n-1}{R}
$

Lens Maker’s Formula (Derived from Spherical Surface Refraction)

For a thin lens made of refracting surfaces:

$
\frac{1}{f}=(n-1)\left(\frac{1}{R_1}-\frac{1}{R_2}\right)
$

Where:
- $f$ : Focal length of the lens.
- $n$ : Refractive index of the lens material.
- $R_1, R_2$ : Radii of curvature of the two surfaces of the lens.

Power of a Thin Lens

The power of a lens is a measure of its ability to converge or diverge light. It is defined as the reciprocal of the focal length $(f)$ of the lens, expressed in meters.

$$
P=\frac{1}{f}
$$

Where:
- $P$ : Power of the lens (in diopters, $D$ ).
- $f$ : Focal length of the lens (in meters).

Lens Maker's Formula

The focal length of a thin lens in air is given by:

$$
\frac{1}{f}=(n-1)\left(\frac{1}{R_1}-\frac{1}{R_2}\right)
$$

Where:
- $n$ : Refractive index of the lens material.
- $R_1$ : Radius of curvature of the first surface.
- $R_2$ : Radius of curvature of the second surface.
- $f$ : Focal length of the lens.

Power in terms of Lens Maker's Formula:

$$
P=100 \times(n-1)\left(\frac{1}{R_1}-\frac{1}{R_2}\right)
$$

(Here $P$ is in diopters if $R_1$ and $R_2$ are in centimeters.)

Combined Power of Lenses

When two thin lenses of powers $P_1$ and $P_2$ are placed close together:

$$
P_{\mathrm{net}}=P_1+P_2
$$

For lenses separated by a distance $d$ :

$$
P_{\text {net }}=P_1+P_2-\frac{d P_1 P_2}{1000}
$$

(Here $d$ is in meters.)

Refraction Through a Prism

When light passes through a prism, it undergoes refraction twice—once at the first surface (entry) and again at the second surface (exit). The light gets deviated, and the process is governed by the laws of refraction.

Key Terms
1. Angle of Prism ( $A$ ): The angle between the two refracting surfaces.
2. Angle of Incidence ( $i$ ): The angle at which the light ray enters the prism.
3. Angle of Refraction ( $r_1, r_2$ ): The angles of refraction at the two surfaces.
4. Angle of Emergence (e): The angle at which the light ray exits the prism.
5. Angle of Deviation $(D)$ : The angle by which the light ray is deviated from its original path.

Relation Between Angles

1. Angle of Deviation ( $D$ ):

$$
D=(i+e)-A
$$

2. Angle of the Prism $(A)$ :

$$
A=r_1+r_2
$$

Minimum Deviation

- When the angle of deviation $(D)$ is minimum, the prism is in the minimum deviation condition.
- At minimum deviation:
- The angle of incidence $(i)$ equals the angle of emergence $(e)$.
- The path of light through the prism is symmetrical.

Formula at Minimum Deviation:

$$
\begin{gathered}
D_m=2 i-A \\
i=\frac{A+D_m}{2}
\end{gathered}
$$

Where:
- $D_m$ : Minimum deviation.
- $i$ : Angle of incidence.

Prism Formula

Using Snell's law at the surfaces and minimum deviation condition, the refractive index ( $n$ ) of the prism material is given by:

$$
n=\frac{\sin \left(\frac{A+D_{\text {m }}}{2}\right)}{\sin \left(\frac{A}{2}\right)}
$$

Where:
- $n$ : Refractive index of the prism.
- $A$ : Angle of the prism.
- $D_m$ : Minimum deviation.

How to Apply Ray Optics Formulas

1. Understand the Problem

• Carefully read the question and identify the given values (e.g., object distance, focal length, refractive indices).
• Identify what needs to be calculated (e.g., image distance, magnification, angle of deviation).

2. Identify the Relevant Formula

• Identify the most relevant formula which is more suitable for that particular problem like mirror formula, lens formula, prism formula etc.

3. Assign Sign Conventions

Use the proper sign conventions for distances and radii:

• For mirrors: Follow the New Cartesian Sign Convention.
• For lenses: Distances measured in the direction of incident light are positive.

4. Substitute the Known Values

• Substitute the given data into the selected formula.
• Ensure the units are consistent (e.g., convert cm to m if needed).

5. Solve the Equation

• Rearrange the equation to isolate the unknown quantity.
• Perform calculations carefully to avoid arithmetic errors.

6. Verify the Results

• Check if the result is physically meaningful (e.g., a positive focal length for a convex lens, negative for a concave lens).
• Analyze the result for accuracy in terms of signs and magnitude.