#wave interference

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 phase difference corresponding to a path difference of \( 5\lambda/4 \) in a double-slit experiment?

**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. Phase difference Φ = (2π/λ) Δ . For Δ = (5λ/4) , Φ = (2π/λ) · (5λ/4) = (5π/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 and A = 2a cos(φ/2), calculation gives (5π/2), illustrating interference, diffraction and polarization principles.

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 intensity at a point in a double-slit experiment where the path difference is \( 3\lambda/4 \), if the maxim

**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. Intensity I = 4I₀ cos²(Φ/2) , where Φ = (2π/λ) Δ . For Δ = (3λ/4) , Φ = (2π/λ) · (3λ/4) = (3π/2) , I = 4I₀ cos²((3π/4)) = 4I₀ ((√(2)/2))² = 4I₀ × (1/2) = 2I₀ . Using Δ = d sinθ, y = n λ D/d, a sinθ = n λ, I = I₀ cos²θ, n = c/v,

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 \( 5\pi \)?

**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 = 5π , A = 2a cos((5π/2)) = 2a × 0 = 0 . 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 0, illustrating interference, diffraction and polarization principles.

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 \( 3\pi \)?

**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 = 3π , A = 2a cos((3π/2)) = 2a × 0 = 0 . 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 0, illustrating interference, diffraction and polarization principles.

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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Superposition, Resultant Amplitude and Intensity

What is the phase difference between two coherent waves resulting in destructive interference?

**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. Destructive interference occurs when the phase difference is an odd multiple of π , with the simplest case being Φ = π . 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 π, illustrating interference, diffraction and pola

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

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Superposition, Resultant Amplitude and Intensity

What is the amplitude of the resultant wave when two coherent waves of amplitude \( a \) interfere with a phase differen

**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 = π/2 , A = 2a cos(π/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 and A = 2a cos(φ/2), calculation gives a√(2), illustrating interference, diffraction and polarization principles.

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

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Displacement Current and Ampere-Maxwell Law

In electromagnetic theory, what explains the ability of waves to exhibit coherence?

**Displacement current** I_d = ε₀ dΦ_E/dt, Φ_E = ∫ E·dA electric flux (V·m), ε₀=8.85×10⁻¹² F/m, ensures continuity of current in charging capacitor where conduction current stops between plates, I_d equals conduction current in wires, 3 A conduction ⇒ 3 A displacement, maintaining Ampere's law ∮ B·dl = μ₀(I_c+I_d). Coherence arises from the wave-like nature of electromagnetic waves, where consistent phase relationships between oscillations allow phenomena like interference to occur. Using c = fλ, E₀/B₀ = c, I_d = ε₀ dΦ_E/dt, and spectrum classification λ = c/f, evaluation yields Wave-like natu

Ref: NCERT > Physics Book > Electromagnetic Waves > Displacement Current and Ampere-Maxwell Law

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