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The latest Messages 2
2021-08-28 20:13:45
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883 views17:13
2021-08-28 09:24:11
Notes on Wave Optics
Huygens Principle:- Wave-front of a wave, at any instant , is defined as the locus of all the particles in the medium which are being disturbed at the same instant of time and are in the same phase of vibration.
(a) Each point on a wave front acts as a source of new disturbance and emits its own set of spherical waves called secondary wavelets. The secondary wavelets travel in all directions with the velocity of light so long as they move in the same medium.
(b) The envelope or the locus of these wavelets in the forward direction gives the position of new wave front at any subsequent time.
Determination of Phase Difference:-
The phase difference between two waves at a point will depend upon
(a) The difference in path lengths of the two waves from their respective sources.
(b) The refractive index of the medium
(c) Initial phase difference between the source if any.
(d) Reflections, if any, in the path followed by waves.
Reflection of plane wave at plane surface (Laws of reflection):-
a) The incident ray, the reflected ray and normal to the reflecting surface at the point of incidence, all lie in one plane and that plane is perpendicular to the reflecting surface.
b) The angle of incidence is equal to the angle of reflection.
So, ∠i = ∠r
This signifies angle of incidence is equal to the angle of reflection.
Refraction of light:-
Refraction is the phenomena by virtue of which a wave going from one medium to another undergoes a change in velocity.
(a) The sine of the angle between the incident ray and the normal bears a constant ratio to the sine of the angle between refracted ray and the normal.
sin i/sin r = v1/v2 = 1µ2 = constant
Here, v1 and v2 are the velocities of sound in first and second medium respectively.1µ2 is the refractive index of the second medium with respect to first.
(b) The incident ray, the refracted ray and the normal to the refracting surface lie in the same plane.
Interference:- The modification in the distribution of light energy obtained by the superposition of two or more waves is called interference.
Principle of superposition:- It states that a number of waves travelling, simultaneously, in a medium behave independent of each other and the net displacement of the particle, at any instant, is equal to the sum of the individual displacements due to all the waves.
Displacement equation:- y = R sin 2π/λ (vt+x/2)
Amplitude:- R = 2a cos πx/λ
Intensity:- I = K4a2 cos2 (πx/λ) [I = KR2]
Maxima:- A point having maximum intensity is called maxima.
x = 2n (λ/2)
A point will be a maxima if the two waves reaching there have a path difference of even multiple of λ/2.
Imax = 4Ka2 = 4i (Here, i = Ka2)
Minima:- A point having minimum intensity is called a minima.
x = (2n+1) (λ/2)
A point will be a minima if the two waves reaching there have a path difference of odd multiple of λ/2.
Imin = K. 4a2×0 = 0
Condition for constructive interference:-
Path difference = (2n)λ/2
Phase difference = (2n)π
Condition for destructive interference:-
Path difference = (2n+1)λ/2
Phase difference = (2n+1)π
Coherent Sources:- Coherent sources are the sources which either have no phase difference or have a constant difference of phase between them.
Conditions for interference:-
(a) The two sources should emit, continuously, waves of same wavelength or frequency.
(b) The amplitudes of the two waves should be either or nearly equal
(c) The two sources should be narrow.
(d) The sources should be close to each other.
(e) The two sources should be coherent one.
Young’s double slit experiment:-
Path difference, x = yd/D
Maxima, y = nλD/d
Here, n = 0,1,2,3….
Minima, y = (2n+1) λD/d
Here, n = 0,1,2,3….
Fringe Width:- It is the distance between two consecutive bright and dark fringes.
β = λD/d
Displacement of fringes due to the introduction of a thin transparent medium:-
(a) Shift for a particular order of fringes:-
?y = (β/λ) (µ-1)t
(b) Shift across a particular point of observation:-
µ = (mλ/t) +1
Lloyd’s single mirror:-
?λ = β .2a/D
Power of lens:- P = 100/f
4.8K views06:24
2021-08-27 18:43:10
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9.1K views15:43
2021-08-27 16:43:11
Notes on Trigonometric Equations and Identities
A function f(x) is said to be periodic if there exists some T > 0 such that f(x+T) = f(x) for all x in the domain of f(x).
In case, the T in the definition of period of f(x) is the smallest positive real number then this ‘T’ is called the period of f(x).
Periods of various trigonometric functions are listed below:
1) sin x has period 2π
2) cos x has period 2π
3) tan x has period π
4) sin(ax+b), cos (ax+b), sec(ax+b), cosec (ax+b) all are of period 2π/a
5) tan (ax+b) and cot (ax+b) have π/a as their period
6) |sin (ax+b)|, |cos (ax+b)|, |sec(ax+b)|, |cosec (ax+b)| all are of period π/a
7) |tan (ax+b)| and |cot (ax+b)| have π/2a as their period
Sum and Difference Formulae of Trigonometric Ratios
1) sin(a + ß) = sin(a)cos(ß) + cos(a)sin(ß)
2) sin(a – ß) = sin(a)cos(ß) – cos(a)sin(ß)
3) cos(a + ß) = cos(a)cos(ß) – sin(a)sin(ß)
4) cos(a – ß) = cos(a)cos(ß) + sin(a)sin(ß)
5) tan(a + ß) = [tan(a) + tan (ß)]/ [1 - tan(a)tan (ß)]
6)tan(a - ß) = [tan(a) - tan (ß)]/ [1 + tan (a) tan (ß)]
7) tan (π/4 + θ) = (1 + tan θ)/(1 - tan θ)
8) tan (π/4 - θ) = (1 - tan θ)/(1 + tan θ)
9) cot (a + ß) = [cot(a) . cot (ß) - 1]/ [cot (a) +cot (ß)]
10) cot (a - ß) = [cot(a) . cot (ß) + 1]/ [cot (ß) - cot (a)]
Double or Triple -Angle Identities
1) sin 2x = 2sin x cos x
2) cos2x = cos2x – sin2x = 1 – 2sin2x = 2cos2x – 1
3) tan 2x = 2 tan x / (1-tan 2x)
4) sin 3x = 3 sin x – 4 sin3x
5) cos3x = 4 cos3x – 3 cosx
6) tan 3x = (3 tan x - tan3x) / (1- 3tan 2x)
For angles A, B and C, we have
1) sin (A + B +C) = sinAcosBcosC + cosAsinBcosC + cosAcosBsinC - sinAsinBsinC
2) cos (A + B +C) = cosAcosBcosC- cosAsinBsinC - sinAcosBsinC - sinAsinBcosC
3) tan (A + B +C) = [tan A + tan B + tan C –tan A tan B tan C]/ [1- tan Atan B - tan B tan C –tan A tan C
4) cot (A + B +C) = [cot A cot B cot C – cotA - cot B - cot C]/ [cot A cot B + cot Bcot C + cot A cotC–1]
List of some other trigonometric formulas:
1) 2sinAcosB = sin(A + B) + sin (A - B)
2) 2cosAsinB = sin(A + B) - sin (A - B)
3) 2cosAcosB = cos(A + B) + cos(A - B)
4) 2sinAsinB = cos(A - B) - cos (A + B)
5) sin A + sin B = 2 sin [(A+B)/2] cos [(A-B)/2]
6) sin A - sin B = 2 sin [(A-B)/2] cos [(A+B)/2]
7) cosA + cos B = 2 cos [(A+B)/2] cos [(A-B)/2]
8) cosA - cos B = 2 sin [(A+B)/2] sin [(B-A)/2]
9) tanA ± tanB = sin (A ± B)/ cos A cos B
10)cot A ± cot B = sin (B ± A)/ sin A sin B
Method of solving a trigonometric equation:
1) If possible, reduce the equation in terms of any one variable, preferably x. Then solve the equation as you used to in case of a single variable.
2) Try to derive the linear/algebraic simultaneous equations from the given trigonometric equations and solve them as algebraic simultaneous equations.
3) At times, you might be required to make certain substitutions. It would be beneficial when the system has only two trigonometric functions.
Some results which are useful for solving trigonometric equations:
1) sin θ = sina and cosθ = cosa ⇒ θ = 2nπ + a
2) sin θ = 0 ⇒ θ = nπ
3) cosθ = 0 ⇒ θ = (2n + 1)π/2
4) tan θ = 0 ⇒ θ = nπ
5) sinθ = sina⇒ θ = nπ + (-1)na where a ∈ [–π/2, π/2]
6) cosθ= cos a ⇒ θ = 2nπ ± a, where a ∈[0,π]
7) tanθ = tana⇒ θ = nπ+ a, where a ∈[–π/2, π/2]
8) sinθ = 1 ⇒ θ= (4n + 1)π/2
9) sin θ = -1 ⇒ θ = (4n - 1) π /2
10) sin θ = -1 ⇒ θ = (2n +1) π /2
11) |sinθ| = 1⇒ θ =2nπ
12) cosθ = 1 ⇒ θ =(2n + 1)
13) |cosθ| = 1⇒ θ =nπ
@arvind_arora_vedantu_official
11.2K viewsedited 13:43
2021-08-26 19:34:11
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3.8K views16:34
2021-08-25 13:04:54
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2021-08-25 11:31:52
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4.4K views08:31
2021-08-24 18:34:40
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