**Important Derivations for Class 12 Physics Chapter 10 Wave Optics**

**Derivations related to Huygens’ Principle**

(1) **Using Huygensâ€™ principle, verify the laws of reflection at a plane surface.**

First the wavefront touches the reflecting surface at B and then at the successive points towards C. In accordance with Huygensâ€™ principle, from each point on BC secondary wavelets start growing with the speed c. During the time the disturbance from A reaches the point C, the secondary wavelets from B must have spread over a hemisphere of radius DB = AC = ct, where ‘t’ is the time taken by the disturbance to travel from A to C. The tangent plane CD drawn from the point ‘C’ over this hemisphere of radius ‘ct’ will be the new reflected wavefront. Let angles of incidence and reflection be i and r respectively. In Î”ABC and Î”DCB, we have

$$ \begin{array}{ll} \angle B A C=\angle C D B & {\left[\text { Each is } 90^{\circ}\right]} \\ B C=B C & \text { [Common] } \\ A C=B D & \text { [Each is equal to } c t] \\ â‡’\quad \triangle A B C \cong \triangle D C B & \\ \text { Hence } \angle A B C=\angle D C B & \\ \text { or } \quad i=r & \end{array} $$i.e., the angle of incidence is equal to the angle of reflection. This proves the first law of reflection.

(2) **Use Huygensâ€™ principle to verify the laws of refraction.**

Laws of refraction: Suppose when distribution from point P on incident wave front reaches point P on the refracted wave front the disturbance from point Q reaches the point Q or the refracting surface XY. Since, A â€˜Qâ€™ Pâ€™ represents the refracted wave front the time takes by light to travel from a point on incident wave front to the corresponding point on refracted wave front would always be the same. Now, time taken by light to go from r to Qâ€™ will be-

$$ t=\frac{Q k}{c}+\frac{k Q^{\prime}}{v} \ldots(\mathrm{i}) $$ (Where c and v are the velocities of light in two medium)In right angled $$\triangle A Q k, \Delta Q A k=I,$$ $$Q k=A k \sin i \quad \ldots(ii)$$ In right angle $$\Delta P^{\prime} Q^{\prime}, \Delta Q P^{\prime} Q^{\prime} k=r$$ $$K Q^{\prime}=\mathrm{KP}^{\prime} \sin \mathrm{r} \ldots(iii)$$ Substituting eq. (ii) and (iii) in eq. (i), we get $$ \begin{aligned} & t=\frac{A k \sin i}{c}+\frac{K P^{\prime} \sin r}{\nu} \\ & \text { or } t=\frac{A k \sin i}{c}+\frac{\left(A P^{\prime}-A k\right) \sin r}{\nu} \end{aligned} $$

The rays from different points or the incident wave front will take the same time to reach the corresponding points on the refracted wave front i.e., given by equation (iv) is independent of Ak. will happens so, if

$$ \begin{aligned} & \frac{\sin i}{c}-\frac{\sin r}{v}=0 \\ & =\frac{\sin i}{\sin r}=\frac{c}{v} \end{aligned} $$ However, $$\frac{c}{v}=n$$This is the ** shellâ€™s law** for refraction of light.

## Why students face difficulty in physics derivations?

There can be several reasons why students face difficulty in physics derivations, including:

- Lack of foundation: A lack of understanding of the underlying concepts and principles can make it difficult to follow the logical steps in a derivation.
- Mathematical background: Physics often involves complex mathematical calculations, and students who struggle with math may find it challenging to perform the necessary calculations.
- Limited practice: Regular practice is key to developing the skills required for physics derivations, and students who have limited practice opportunities may struggle.
- Poor problem-solving skills: Physics problems often require creative problem-solving skills, and students who struggle with this aspect of physics may find derivations particularly challenging.
- Limited exposure to different problems: Physics derivations can vary widely in their level of complexity, and students who have limited exposure to different types of problems may struggle when faced with a new challenge.
- Confusion with notation: Physics often uses a specialized notation, and students who are unfamiliar with this notation may struggle to follow the steps in a derivation.

## Remembering physics derivations can be a challenge, but here are some tips that may help:

- Practice, practice, practice: Regularly practicing physics derivations helps to build muscle memory and improve recall.
- Visualize the process: Try to visualize the steps involved in a derivation, as well as the physical meanings behind each equation.
- Make connections: Try to relate each step of the derivation to a concept or formula you already know, which can help to reinforce your understanding.
- Write it down: Writing out a derivation helps to solidify your understanding and makes it easier to remember.
- Understand the physical meaning: Try to understand the physical meaning behind each equation, which can help you to remember the derivation in a broader context.
- Teach someone else: Teaching someone else the derivation can be a great way to reinforce your understanding and remember it more easily.
- Use mnemonics: Creating mnemonics or acronyms can be a helpful way to remember a series of steps in a derivation.

Remember, it takes time and consistent effort to remember physics derivations. Keep practicing and seeking help when needed, and you will likely see improvement over time.