A beam of light converges at a point $P$. Now a lens is placed in the path of the convergent beam $12 \mathrm{c}\mathrm{m}$ from $P$. At what point does the beam converge if the lens is a convex lens of focal length $20 \mathrm{c}\mathrm{m}$.

(i) Reflection is governed by the equation $\mathrm{\angle }\mathrm{i}=\mathrm{\angle }\mathrm{r}$. Angles of incidence and reflection are $\mathrm{i}$ and $\mathrm{r}$ respectively.

(ii) The incident ray, reflected ray and normal lie in the same plane.

2. Refraction of light:

(i) Refraction is given by the Snell’s law, $\frac{\sin \mathrm{i}}{\sin \mathrm{r}}=\mathrm{n}$, Angles of incidence, refraction and relative refractive index of medium are $\mathrm{i}$,$\mathrm{r}$ and $\mathrm{n}$ respectively.

(ii) The incident ray, refracted ray and normal lie in the same plane.

3. Critical Angle and Total Internal Reflection:

(i) Total internal reflection is complete reflection of light, when it travels from denser medium to rarer medium.

(ii) The critical angle of incidence ${\mathrm{i}}_{\mathrm{c}}$ for a ray incident from a denser to rarer medium, is that angle for which the angle of refraction is 90°. For $\mathrm{i}>{\mathrm{i}}_{\mathrm{c}}$, total internal reflection occurs.

4. Mirror Equation and Cartesian Sign Convention:

(i) Cartesian sign convention: Distances measured in the same direction as the incident light are positive; those measured in the opposite direction are negative. The heights measured upwards above x-axis are taken as positive and downwards are taken as negative.

(ii) Mirror equation: $\frac{1}{\mathrm{v}}+\frac{1}{\mathrm{u}}=\frac{1}{\mathrm{f}}$, where $\mathrm{u}$ and $\mathrm{v}$ are object and image distances, respectively, and $\mathrm{f}$ is the focal length of the mirror. $\mathrm{f}$ is (approximately) half the radius of curvature $\mathrm{R}$.

(iii) Focal lenght $\left(f\right)$ is negative for concave mirror and it is positive for a convex mirror.

5. Minimum Deviation through a Prism:

(i) For a prism of the angle A, of refractive index n2 placed in a medium of refractive index n1, $n_{21}=\frac{n_2}{n_1}=\frac{\sin \left[\left(A+{\delta }_m\right)/2\right]}{\sin (A/2)}$ where ${\delta }_m$ is the angle of minimum deviation.

(ii) At the minimum deviation condition, angle of incidence is equal to the angle if emergence.

6. Refraction through Spherical Surface and Lens:

(i) For refraction through a spherical interface (from medium 1 to 2 of refractive index ${\mathrm{n}}_1$and ${\mathrm{n}}_2$ respectively) $\frac{{\mathrm{n}}_2}{\mathrm{v}}-\frac{{\mathrm{n}}_1}{\mathrm{u}}=\frac{{\mathrm{n}}_2-{\mathrm{n}}_1}{\mathrm{R}}$

(ii) Thins lens formula: $\frac{1}{\mathrm{v}}-\frac{1}{\mathrm{u}}=\frac{1}{\mathrm{f}}$

(iii) Lens maker’s formula: $\frac{1}{\mathrm{f}}=\frac{\left({\mathrm{n}}_2-{\mathrm{n}}_1\right)}{{\mathrm{n}}_1}\left(\frac{1}{{\mathrm{R}}_1}-\frac{1}{{\mathrm{R}}_2}\right)$ where ${\mathrm{R}}_1$ and ${\mathrm{R}}_2$ are the radii of curvature of the lens surfaces. $\mathrm{f}$ is positive for a converging lens; $\mathrm{f}$ is negative for a diverging lens.

(iv) The power of a lens $\mathrm{P}=1/\mathrm{f}$. The SI unit for power of a lens is Diopter D:1 D=1m-1

(v) If several thin lenses of focal length ${\mathrm{f}}_1,{\mathrm{f}}_2,{\mathrm{f}}_3,\dots$ are in contact, the effective focal length of their combination, is given by $\frac{1}{\mathrm{f}}=\frac{1}{{\mathrm{f}}_1}+\frac{1}{{\mathrm{f}}_2}+\frac{1}{{\mathrm{f}}_3}+\cdots$    and the total power of a combination of several lenses is $\mathrm{P}={\mathrm{P}}_1+{\mathrm{P}}_2+{\mathrm{P}}_3+\dots$

7. Dispersion of light:

(i) Dispersion is the splitting of light into its constituent colours.

(ii) Refractive index of the medium depends on the frequency of the light.

8. Eye and its Defects:

(i) The focal length of the eye-lens can be varied such that the image is always formed on the retina by the ability of the eye called as accommodation.

(ii) In a defective eye, if the image is focused before the retina (myopia), a diverging corrective lens is needed; if the image is focused beyond the retina (hypermetropia), a converging corrective lens is needed. Astigmatism is corrected by using cylindrical lenses.

9. Optical Instruments:

(i) When the image is formed at near point, the magnifying power $\mathrm{m}$ of a simple microscope is given by $\mathrm{m}=1+(\mathrm{D}/\mathrm{f})$ where $\mathrm{D}=25\mathrm{ }\mathrm{cm}$ is the least distance of distinct vision, and f is the focal length of the convex lens. If the image is formed at infinity, $\mathrm{m}=\mathrm{D}/\mathrm{f}$.

(ii) For a compound microscope, the magnifying power is given by $\mathrm{m}={\mathrm{m}}_{\mathrm{e}}\times {\mathrm{m}}_0$, where me ${\mathrm{m}}_{\mathrm{e}}=1+\left(\mathrm{D}/{\mathrm{f}}_{\mathrm{e}}\right)$ is the magnification due to the eyepiece, and mo is the magnification produced by the objective.

(iii) For a compound microscope, the magnifying power is given by, approximately, $\mathrm{m}=\frac{\mathrm{L}}{{\mathrm{f}}_{\mathrm{o}}}\times \frac{\mathrm{D}}{{\mathrm{f}}_{\mathrm{e}}}$ when the image is formed at infinity, and $\mathrm{m}=\frac{\mathrm{L}}{{\mathrm{f}}_{\mathrm{o}}}\times \left(1+\frac{\mathrm{D}}{{\mathrm{f}}_{\mathrm{e}}}\right)$ when the image is formed at near point, where ${\mathrm{f}}_0$ and ${\mathrm{f}}_{\mathrm{e}}$ are the focal lengths of the objective and eyepiece, respectively, and $\mathrm{L}$ is the distance between their focal points.

(iv) Magnifying power $\mathrm{m}$ of a telescope is the ratio of the angle $\mathrm{\beta }$ subtended at the eye by the image to the angle $\mathrm{\alpha }$ subtended at the eye by the object. $\mathrm{m}=\frac{\mathrm{\beta }}{\mathrm{\alpha }}=\frac{{\mathrm{f}}_{\mathrm{o}}}{{\mathrm{f}}_{\mathrm{e}}}$, where ${\mathrm{f}}_0$ and ${\mathrm{f}}_{\mathrm{e}}$ are the focal lengths of the objective and eyepiece, respectively.