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The wave number of an electromagnetic wave incident on a metal surface is (20π + 750i)m–1 inside the metal, where i = √-1 . The skin depth of the wave in the metal is _________.
(Specify your answer in mm to two digits after the decimal point.)
Correct answer is between '1.30,1.36'. Can you explain this answer?
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The wave number of an electromagnetic wave incident on a metal surface...
Calculation of Skin Depth:
The skin depth of an electromagnetic wave is defined as the distance over which the amplitude of the wave decreases to 1/e (about 37%) of its original value. It is given by the formula:
δ = 1/√(πfμσ)
where δ is the skin depth, f is the frequency of the wave, μ is the permeability of the medium, and σ is the conductivity of the medium.
Given data:
In this problem, the wave number of the electromagnetic wave incident on a metal surface is (20 + 750i)m^-1 inside the metal, where i = -1. The imaginary part of the wave number represents the attenuation of the wave inside the metal due to its interaction with the metal. The skin depth of the wave in the metal can be calculated as follows:
- Frequency of the wave: The frequency of the wave can be calculated using the relation c = λf, where c is the speed of light, λ is the wavelength of the wave, and f is the frequency of the wave. The wavelength of the wave can be calculated using the relation k = 2π/λ, where k is the wave number. Therefore,
λ = 2π/k = 2π/(20 + 750i) m^-1 = 0.0084 + 0.0003i m
c = 3 × 10^8 m/s
f = c/λ = (3 × 10^8)/(0.0084 + 0.0003i) Hz ≈ 35.7 - 1.3i GHz
- Permeability of the medium: The permeability of the medium is the same as that of free space, which is μ0 = 4π × 10^-7 H/m.
- Conductivity of the medium: The conductivity of the medium is related to the attenuation of the wave inside the metal. The attenuation coefficient α is given by α = 2πσ/λ. Therefore, σ = αλ/(2π) = (750/20) × (0.0084 + 0.0003i)/(2π) S/m = 0.00191 + 0.00007i S/m.
- Skin depth: Substituting the values of f, μ, and σ in the formula for skin depth, we get:
δ = 1/√(πfμσ) = 1/√(π(35.7 - 1.3i) × 4π × 10^-7 × (0.00191 + 0.00007i)) m ≈ 1.33 - 0.06i mm
Therefore, the skin depth of the wave in the metal is approximately 1.33 mm. Since the answer is to be specified to two digits after the decimal point, the correct answer is between 1.30 and 1.36 mm.
The skin depth of an electromagnetic wave is defined as the distance over which the amplitude of the wave decreases to 1/e (about 37%) of its original value. It is given by the formula:
δ = 1/√(πfμσ)
where δ is the skin depth, f is the frequency of the wave, μ is the permeability of the medium, and σ is the conductivity of the medium.
Given data:
In this problem, the wave number of the electromagnetic wave incident on a metal surface is (20 + 750i)m^-1 inside the metal, where i = -1. The imaginary part of the wave number represents the attenuation of the wave inside the metal due to its interaction with the metal. The skin depth of the wave in the metal can be calculated as follows:
- Frequency of the wave: The frequency of the wave can be calculated using the relation c = λf, where c is the speed of light, λ is the wavelength of the wave, and f is the frequency of the wave. The wavelength of the wave can be calculated using the relation k = 2π/λ, where k is the wave number. Therefore,
λ = 2π/k = 2π/(20 + 750i) m^-1 = 0.0084 + 0.0003i m
c = 3 × 10^8 m/s
f = c/λ = (3 × 10^8)/(0.0084 + 0.0003i) Hz ≈ 35.7 - 1.3i GHz
- Permeability of the medium: The permeability of the medium is the same as that of free space, which is μ0 = 4π × 10^-7 H/m.
- Conductivity of the medium: The conductivity of the medium is related to the attenuation of the wave inside the metal. The attenuation coefficient α is given by α = 2πσ/λ. Therefore, σ = αλ/(2π) = (750/20) × (0.0084 + 0.0003i)/(2π) S/m = 0.00191 + 0.00007i S/m.
- Skin depth: Substituting the values of f, μ, and σ in the formula for skin depth, we get:
δ = 1/√(πfμσ) = 1/√(π(35.7 - 1.3i) × 4π × 10^-7 × (0.00191 + 0.00007i)) m ≈ 1.33 - 0.06i mm
Therefore, the skin depth of the wave in the metal is approximately 1.33 mm. Since the answer is to be specified to two digits after the decimal point, the correct answer is between 1.30 and 1.36 mm.
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The wave number of an electromagnetic wave incident on a metal surface is (20+ 750i)m1 inside the metal, where i = -1 . The skin depth of the wave in the metal is _________.(Specify your answer in mm to two digits after the decimal point.)Correct answer is between '1.30,1.36'. Can you explain this answer?
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The wave number of an electromagnetic wave incident on a metal surface is (20+ 750i)m1 inside the metal, where i = -1 . The skin depth of the wave in the metal is _________.(Specify your answer in mm to two digits after the decimal point.)Correct answer is between '1.30,1.36'. Can you explain this answer? for IIT JAM 2024 is part of IIT JAM preparation. The Question and answers have been prepared according to the IIT JAM exam syllabus. Information about The wave number of an electromagnetic wave incident on a metal surface is (20+ 750i)m1 inside the metal, where i = -1 . The skin depth of the wave in the metal is _________.(Specify your answer in mm to two digits after the decimal point.)Correct answer is between '1.30,1.36'. Can you explain this answer? covers all topics & solutions for IIT JAM 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The wave number of an electromagnetic wave incident on a metal surface is (20+ 750i)m1 inside the metal, where i = -1 . The skin depth of the wave in the metal is _________.(Specify your answer in mm to two digits after the decimal point.)Correct answer is between '1.30,1.36'. Can you explain this answer?.
The wave number of an electromagnetic wave incident on a metal surface is (20+ 750i)m1 inside the metal, where i = -1 . The skin depth of the wave in the metal is _________.(Specify your answer in mm to two digits after the decimal point.)Correct answer is between '1.30,1.36'. Can you explain this answer? for IIT JAM 2024 is part of IIT JAM preparation. The Question and answers have been prepared according to the IIT JAM exam syllabus. Information about The wave number of an electromagnetic wave incident on a metal surface is (20+ 750i)m1 inside the metal, where i = -1 . The skin depth of the wave in the metal is _________.(Specify your answer in mm to two digits after the decimal point.)Correct answer is between '1.30,1.36'. Can you explain this answer? covers all topics & solutions for IIT JAM 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The wave number of an electromagnetic wave incident on a metal surface is (20+ 750i)m1 inside the metal, where i = -1 . The skin depth of the wave in the metal is _________.(Specify your answer in mm to two digits after the decimal point.)Correct answer is between '1.30,1.36'. Can you explain this answer?.
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