Question

derivation of snell's law .
explain in deatil

Answer 1

__________________________

$\; \; \; \; \; \; \; \; \;{\large{\bold{\sf{\underbrace{Topic \: - \: Refractive \: of \: light \: at \: plane \: surface}}}}}$

__________________________

Actually frankly saying that the Snell's law is derived in so much ways.

➨ The Snell's law can be derived from Fermat's principal.

➨ Fermat's principal means or state that the light travel the path which take the least time.

➨ The optical path length the stationery point is find while giving the path taken by light.

${\sf{\underline{Derivation \; of \; snell's \; law}}}$

As we already know that the refractive index μ is given by ${\bold{\sf{n = \dfrac{c}{v}}}}$

Where,

c denotes speed of light in air.

v denotes speed of light in medium

➝ ${\bold{\sf{t = \dfrac{\sqrt{ {x}^{2} + h ^{2 _{1}}}}{c / n_{1}}}}}$ + ${\bold{\sf{ \dfrac{(l - x)^{2} + h^{2}_{2}}{c / n_{2}}}}}$

➝ ${\bold{\sf{\dfrac{dt}{dx}}}}$ = ${\bold{\sf{\dfrac{n_{1}x}{c \sqrt{{x}^{2} + h^{2}_{1}}}}}}$ + ${\bold{\sf{ \dfrac{ -n_{2}(l - x)}{c \sqrt{(l - x)^{2} + h^{2}_{2}}}}}}$ = 0

➝ ${\bold{\sf{\dfrac{n_{1}x}{n_{2}(l - x) + h ^{2}_{2}}}}}$ = ${\bold{\sf{ \dfrac{n_{2} (l - x)}{ \sqrt{(l - x)^{2} + h^{2}_{2}}}}}}$

➝ n₁sinθ₁ = n₂sinθ₂

${\sf{\underline{Explanation}}}$

According to the given attachment let's see consider a ray of light is travelling from point A in a medium with μ to point B in a medium n1 here to the point B in a medium with μ n2. The time travelled between that 2 points is the distance in each medium divided by the light's speed in medium.

To minimise the time we set the derivatives of time with respect to point x = 0. We also use definition of sine as the opposite side over hypotenuse to relates the length to the angles of incidence and angle of refraction.

__________________________

$\; \; \; \; \; \; \; \; \;{\large{\bold{\sf{\underbrace{More \; information}}}}}$

__________________________

${\sf{\underline{What \; is \; light \: refraction}}}$

⇢ When a ray of light propagation in a medium enters the other medium, it deviates from it's path. This phenomena of change in the direction of propagation of light at the boundary when it's passes from one medium to another medium is known to be "Refractive of light."

${\sf{\underline{Law \; of \; refraction}}}$

❶ Incident ray, refracted ray and normal drawn at incident point always lie in the same plane.

❷ Snell's law = For a given colour of light, the ratio of sine of angle of incidence to the sine of angle of refraction is a constant, i.e., sin i / sin r = ¹μ₂ ( it's constant )

This constant ¹μ₂ is called refractive index of second medium with respect to the first medium.

sinθ₂ / sinθ = v / v = n /n

Where,

➨ θ is the angle measure from the normal of the boundary.

➨ v is velocity of light in respective medium.

➨ n is refractive index in respective medium.

⇢ When a ray of light enters from one medium to to another media its frequency and phase don't change but wavelength and velocity changes.

⇢ Absolute refractive index of a medium is defined as the ratio of speed of light in free space to that is given medium i.e., absolute refractive index (μ) = ${\bold{\sf{\dfrac{Light \: speed \: in \: vaccum}{Light \: speed \: in \: medium}}}}$

⇢ The refractive index of a medium is different for different colours. μ of a medium decreased with increased in wavelength' of light. Henceforth, μ of a medium is maximum for violet colour of light and minimum for red colour of light.

⇢ The μ of a medium decrease with the increase in temperature. But this is very small variation.

${\sf{\underline{Refractive \; index}}}$

⇢ Refractive index of a medium is defined as thr ratio of speed of light in vaccum to speed of light in a medium

${\bold{\sf{\dfrac{c}{v} = \dfrac{Light \: speed \: in \: vaccum}{Light \: speed \: in \: medium}}}}$ = μ

⇢ Speed of light is different in different medium. Velocity of light is larger in medium which is small refractive index.

${\sf{\underline{Light \: speed}}}$

❶ Vaccum = 3 × 10⁸ m/s

❷ Water = 2.25 × 10⁸ m/s

❸ Nylon = 1.96 × 10⁸

❹ Glass = 2 × 10⁸

❺ Salt ( Rock ) = 1.96 × 10⁸

${\sf{\underline{Angle \; of \; I \; and r}}}$

➨ For a smooth surface the angle of incidence equals the angle of reflection as measured with reference to perpendicular line. Angle of reflection is always equal to angle of reflection.

➨ θr = θi

➨ Angle r = Angle i

${\sf{\underline{Relationship \; between \; angle \; of \; I \; and r}}}$

➨ The law of reflection states that, on reflection from a smooth surface, the angle of the reflected ray is equal to the angle of the incident

Answer 2

Given :

Area of rhombus = 120 cm²

Length of diagonal = 8 cm

To find :

Length of another diagonal

According to the question,

$\sf{ : \implies Area \: of \: rhombus = \dfrac{1}{2} \times d _{1} \times d_{2} }$

$\sf : \implies{ {120 \: cm}^{2} = \dfrac{1}{2} \times 8 \: cm \times x}$

$\sf : \implies{ {120 \: cm}^{2} \times 2 = 8 \: cm \times x }$

$\sf : \implies{ {240 \: cm}^{2} = 8 \: cm \times x}$

$\sf : \implies{ \dfrac{240}{8} \: cm = x}$

${ \underline{ \boxed{ \sf \pink{ : \implies{ \bm3 \bm0 \: c m =x}}}}}$

${ \therefore{ \underline{\sf{So \:,the \: length \: of \: other \: diagonal \: is \: 3 0 \: cm}}}}$