#resultant amplitude

Optical Phenomena and Applications

What is the resultant amplitude of two coherent waves of amplitude \( a \) with a phase difference of \( 4\pi \)?

**Incoherent sources** intensity adds I = I₁+I₂, no interference pattern because phase random, two independent sources cannot produce stable interference because phase difference fluctuates rapidly, coherent sources required with constant phase, laser coherent, visibility of fringes requires coherence, degree of coherence determines contrast. Resultant amplitude A = 2a cos(Φ/2) . For Φ = 4π , A = 2a cos(2π) = 2a × 1 = 2a . Using Δ = d sinθ, y = n λ D/d, a sinθ = n λ, I = I₀ cos²θ, n = c/v, sinC = 1/n, λ' = λ/n and A = 2a cos(φ/2), calculation gives 2a, illustrating interference, diffraction an

Ref: NCERT > Physics Book > Wave Optics > Optical Phenomena and Applications

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Superposition, Resultant Amplitude and Intensity

What is the resultant amplitude of two coherent waves of amplitude \( a \) with a phase difference of \( 7\pi/2 \)?

**Superposition principle** resultant displacement sum of individual, for two coherent waves amplitude a each, resultant amplitude A = √(a² + a² +2a² cosφ)=2a|cos(φ/2)|, phase difference φ, path difference Δ = (φ/2π)λ, for φ=π/2 A=√2 a, for φ=6π cos3π=-1? Actually φ=6π cos3π? A=2a|cos3π|=2a, for φ=4π A=2a, intensity I ∝ A², maximum I_max=4I₀ when φ=0, I=2I₀(1+cosφ)=4I₀ cos²(φ/2). Resultant amplitude A = 2a cos(Φ/2) . For Φ = (7π/2) , A = 2a cos((7π/4)) = 2a ((√(2)/2)) = a√(2) . Using Δ = d sinθ, y = n λ D/d, a sinθ = n λ, I = I₀ cos²θ, n = c/v, sinC = 1/n, λ' = λ/n

Ref: NCERT > Physics Book > Wave Optics > Superposition, Resultant Amplitude and Intensity

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Superposition, Resultant Amplitude and Intensity

What is the resultant amplitude of two coherent waves of amplitude \( a \) with a phase difference of \( 5\pi/2 \)?

**Superposition principle** resultant displacement sum of individual, for two coherent waves amplitude a each, resultant amplitude A = √(a² + a² +2a² cosφ)=2a|cos(φ/2)|, phase difference φ, path difference Δ = (φ/2π)λ, for φ=π/2 A=√2 a, for φ=6π cos3π=-1? Actually φ=6π cos3π? A=2a|cos3π|=2a, for φ=4π A=2a, intensity I ∝ A², maximum I_max=4I₀ when φ=0, I=2I₀(1+cosφ)=4I₀ cos²(φ/2). Resultant amplitude A = 2a cos(Φ/2) . For Φ = (5π/2) , A = 2a cos((5π/4)) = 2a (-(√(2)/2)) = -a√(2) , magnitude a√(2) . Using Δ = d sinθ, y = n λ D/d, a sinθ = n λ, I = I₀ cos²θ, n = c/v, sinC = 1/n,

Ref: NCERT > Physics Book > Wave Optics > Superposition, Resultant Amplitude and Intensity

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Superposition, Resultant Amplitude and Intensity

What is the resultant amplitude of two coherent waves of amplitude \( a \) with a phase difference of \( 2\pi \)?

**Intensity at point** in double-slit I = I_max cos²(φ/2), φ = (2π/λ)Δ, for Δ=λ/4 φ=π/2 I= I_max cos²(π/4)= I_max/2 =2I₀, for Δ=λ/3 φ=2π/3 I= I_max cos²(π/3)= I_max×0.25= I₀, for Δ=5λ/8 φ=5π/4? Actually φ=2π×5/8=5π/4, cos²(5π/8)=?, path difference for destructive φ=(2n+1)π, constructive 2nπ. Resultant amplitude A = 2a cos(Φ/2) . For Φ = 2π , A = 2a cos(π) = 2a × (-1) = -2a , but magnitude is 2a . Using Δ = d sinθ, y = n λ D/d, a sinθ = n λ, I = I₀ cos²θ, n = c/v, sinC = 1/n, λ' = λ/n and A = 2a cos(φ/2), calculation gives 2a,

Ref: NCERT > Physics Book > Wave Optics > Superposition, Resultant Amplitude and Intensity

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Superposition, Resultant Amplitude and Intensity

What is the resultant amplitude of two coherent waves of amplitude \( a \) with a phase difference of \( 6\pi \)?

**Phase difference** corresponding to path difference Δ, φ =2π Δ/λ, for Δ=5λ/8 φ=5π/4, for Δ=9λ/4 φ=9π/2, for Δ=λ path difference φ=2π constructive, but for destructive condition path difference λ can be destructive if one reflection introduces π phase shift, resultant amplitude zero when φ=(2n+1)π. Resultant amplitude A = 2a cos(Φ/2) . For Φ = 6π , A = 2a cos(3π) = 2a × (-1) = -2a , magnitude 2a . Using Δ = d sinθ, y = n λ D/d, a sinθ = n λ, I = I₀ cos²θ, n = c/v, sinC = 1/n, λ' = λ/n and A = 2a cos(φ/2), calculation gives

Ref: NCERT > Physics Book > Wave Optics > Superposition, Resultant Amplitude and Intensity

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Doppler Effect

Two waves \( y_1 = 3 \sin (8x - 16t) \) and \( y_2 = 3 \sin (8x - 16t + \frac{2\pi}{3}) \) interfere. What is the amplit

**Doppler effect** describes apparent frequency shift due to relative motion between source and observer, f' = f·v/(v ∓ v_s) for source motion, f' = f·(v ± v_o)/v for observer motion, upper signs for approach increasing observed frequency. Motion towards observer compresses wavelength raising f'. Amplitude: A = 2a cos (Φ/2) , a = 3 m , Φ = (2π/3) . A = 2 × 3 cos (π/3) = 6 × (1/2) = 3 m . Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields 3 m, illustrating frequency-length-speed interdependence and quantization by boundaries.

Ref: NCERT > Physics Book > Waves > Doppler Effect

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Superposition and Interference of Waves

Two waves \( y_1 = 6 \sin (4x - 12t) \) and \( y_2 = 6 \sin (4x - 12t + \pi) \) interfere. What is the amplitude of the

**Superposition principle** states resultant displacement equals algebraic sum of individual waves, y = y₁ + y₂. For coherent waves with phase difference φ, resultant amplitude A = √(a₁² + a₂² + 2a₁a₂ cosφ), equal amplitudes give A = 2a cos(φ/2), constructive when φ = 2nπ, destructive when φ = (2n+1)π. Amplitude: A = 2a cos (Φ/2) , a = 6 m , Φ = π . A = 2 × 6 cos (π/2) = 12 × 0 = 0 m (destructive interference). Using v = fλ and standing-wave condition fₙ = n v/(2L) or v/(4L) as applicable, calculation yields 0 m, illustrating frequency-length-speed interdependence and quantization by boundarie

Ref: NCERT > Physics Book > Waves > Superposition and Interference of Waves

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