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Questions and Answers
What is the primary focus when discussing the propagation of plane electromagnetic waves in infinite media?
What is the primary focus when discussing the propagation of plane electromagnetic waves in infinite media?
Which mathematical concept is central to solving the wave equation in infinite media?
Which mathematical concept is central to solving the wave equation in infinite media?
In the context of electromagnetics for medical physics, what role does understanding electromagnetic waves play?
In the context of electromagnetics for medical physics, what role does understanding electromagnetic waves play?
What is an example of a plane electromagnetic wave?
What is an example of a plane electromagnetic wave?
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Which parameter is NOT typically considered when analyzing waves in infinite media?
Which parameter is NOT typically considered when analyzing waves in infinite media?
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What do Maxwell's equations primarily describe in the context of plane electromagnetic waves?
What do Maxwell's equations primarily describe in the context of plane electromagnetic waves?
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What is a requirement for the propagation of plane electromagnetic waves in infinite media?
What is a requirement for the propagation of plane electromagnetic waves in infinite media?
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Which of the following statements is NOT true concerning infinite media and electromagnetic wave propagation?
Which of the following statements is NOT true concerning infinite media and electromagnetic wave propagation?
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In the context of plane electromagnetic waves, what does 'uniformity' refer to?
In the context of plane electromagnetic waves, what does 'uniformity' refer to?
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Which equation is essential for understanding the relationship between the electric field and magnetic field in plane electromagnetic waves?
Which equation is essential for understanding the relationship between the electric field and magnetic field in plane electromagnetic waves?
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Study Notes
Propagation of Plane Electromagnetic Waves in Infinite Media
- Electromagnetic waves can transport energy.
- Plane waves in infinite media are the simplest solutions.
- Consider a material with: Β = μΗ, D = E, J = p = 0.
- Maxwell equations, in vector form:
- ∇ × E = - (1/c) (dB/dt)
- ∇ × B = μ₀(J + (1/c)(dE/dt))
- ∇ ⋅ E = ρ/ε₀
- ∇ ⋅ B = 0
Solving the Wave Equation in Infinite Media
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The curl of the curl equations, e.g., Faraday's Law, results in:
- ∇ × (∇ × E) = μ₀ ε₀ (d²E/dt²) / c²
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The divergence of E is zero, so the equation becomes
- μ₀ ε₀ (d²E/dt²) = ∇²(E) / c²
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Identical manipulations with Ampère's law gives a similar equation for B
- μ₀ ε₀ (d²B/dt²) = ∇²(B) / c²
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The wave equation has complex traveling wave solutions:
- Uk(x, t) = ei(kx - wt) where w = vk and k is a real vector
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Derivatives of the complex function:
- ∇Uk = ikUk
- ∇²Uk = - k²Uk
- dUk/dt = -iwUk
- d²Uk/dt² = -w²Uk
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Solutions represent waves traveling in the direction of k.
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Surfaces of constant phase are planes perpendicular to k.
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Velocity of the wave is v = ω/k.
Electromagnetic Fields in Terms of Plane Waves
- Electric and magnetic fields can be represented as:
- E(x, t) = E₀e^(i(kx - wt))
- B(x, t) = B₀e^(i(kx - wt))
- The true fields are the real parts of these complex expressions.
- Maxwell's equations require:
- k ⋅ B₀ = 0
- k ⋅ E₀ = 0
- This means that B₀ and E₀ are perpendicular to k (and thus to the direction of wave propagation).
Conditions Imposed by Maxwell's Equations
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Further conditions on the amplitudes of the fields from other Maxwell equations.
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The electric field amplitude is related to the magnetic field:
- E₀ = (k x B₀) / (iωε₀) or Eo = n x Bo/sqrt(μ₀ε₀)
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The magnetic field is also expressed in terms of the electric field:
- B₀ = (k × E₀) / (iωμ₀) or Bo = √(μ₀ε₀) n x Eo
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Electromagnetic waves are transverse waves (E, B, and k are mutually orthogonal).
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Time-averaged energy density () and Poynting vector (S) are calculated using amplitudes.
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Description
Explore the fundamental principles of electromagnetic wave propagation in infinite media. This quiz covers Maxwell's equations, wave equations, and complex wave solutions, allowing you to grasp how energy is transported through these waves. Test your understanding of these essential concepts in electromagnetic theory.
