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What is the significance of the term ∫𝑣 𝐸⃗.⃗𝐽 𝑑𝑣 in the Poynting theorem?
What is the significance of the term ∫𝑣 𝐸⃗.⃗𝐽 𝑑𝑣 in the Poynting theorem?
- Rate of energy dissipation in the volume V (correct)
- Rate of energy storage in the volume V
- Rate of energy conversion from electric to magnetic field
- Rate of energy transmission from the volume V to the outside
The Poynting theorem is a consequence of Ampere's law.
The Poynting theorem is a consequence of Ampere's law.
True (A)
What is the physical significance of the negative sign in the term 𝜕/𝜕𝑡 ∫𝑣 (𝜀𝐸⃗² + 𝜇𝐻⃗²) 𝑑𝑣?
What is the physical significance of the negative sign in the term 𝜕/𝜕𝑡 ∫𝑣 (𝜀𝐸⃗² + 𝜇𝐻⃗²) 𝑑𝑣?
The negative sign indicates the power being delivered by the field inside the volume to the region outside the volume.
The divergence theorem is used to convert the volume integral of ∇.⃗(𝐸⃗ × 𝐻⃗) to a _______________________ integral.
The divergence theorem is used to convert the volume integral of ∇.⃗(𝐸⃗ × 𝐻⃗) to a _______________________ integral.
What is the name of the theorem that states the rate of energy dissipation in a volume is equal to the rate of energy storage in the volume plus the rate of energy entering the volume from outside?
What is the name of the theorem that states the rate of energy dissipation in a volume is equal to the rate of energy storage in the volume plus the rate of energy entering the volume from outside?
The displacement current is a consequence of the electromagnetic induction.
The displacement current is a consequence of the electromagnetic induction.
What is the physical significance of the term 𝜕/𝜕𝑡 ∫𝑣 (𝜀𝐸⃗² + 𝜇𝐻⃗²) 𝑑𝑣 in the Poynting theorem?
What is the physical significance of the term 𝜕/𝜕𝑡 ∫𝑣 (𝜀𝐸⃗² + 𝜇𝐻⃗²) 𝑑𝑣 in the Poynting theorem?
Match the following physical quantities with their corresponding mathematical representations:
Match the following physical quantities with their corresponding mathematical representations:
The Poynting theorem is a statement of the conservation of _______________________.
The Poynting theorem is a statement of the conservation of _______________________.
The Poynting theorem is a consequence of Maxwell's equations.
The Poynting theorem is a consequence of Maxwell's equations.
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Study Notes
Electromagnetic Induction
- Faraday's law of electromagnetic induction: ∫𝐸⃗ ⋅ 𝑑𝑙 = −∫𝑠 (𝑑𝐵⃗/𝑑𝑡) ⋅ 𝑑𝑆
- The law states that a changing magnetic field induces an electric field
Modified Ampere's Law
- ∇⃗ × 𝐸⃗ = −𝜕𝐵⃗/𝜕𝑡
- The law describes the relationship between electric and magnetic fields
Magnetic Fields
- 𝐵⃗ = 𝜇₀𝐻⃗ - 𝑀
- 𝐻⃗ is the magnetic field strength, 𝐵⃗ is the magnetic flux density, and 𝑀 is the magnetization
Displacement Current Density
- 𝐷⃗ = 𝜖𝐸⃗
- The displacement current density is proportional to the variation of electric field (E)
- It does the same work as the conductive current density 𝐽⃗
Total Current Density
- The total current density is the sum of conductive current density 𝐽⃗ and displacement current density 𝜕𝐷⃗/𝜕𝑡
- The modified Ampere's law becomes: ∇⃗ × 𝐻⃗ = 𝐽⃗ + 𝜕𝐷⃗/𝜕𝑡
Units of Physical Quantities
- 𝐷: C/m²
- 𝐸: V/m
- 𝐵: Tesla or V·s/m²
- 𝐻: A/m
- 𝜌: C/m³
- 𝐽: A/m²
Physical Significance of Maxwell's Equations
- The equations describe the interplay between electric and magnetic fields
- They describe the conservation of energy and the flow of energy between fields
Poynting Theorem
- The theorem states that the rate of energy dissipation in a volume is equal to the rate of decrease of stored energy plus the rate of energy entering the volume from outside
- ∫𝐸⃗ ⋅ 𝐽⃗ 𝑑𝑣 = −(𝜕/𝜕𝑡) ∫(𝜖𝐸⃗² + 𝜇𝐻⃗²) 𝑑𝑣 - ∫(𝐸⃗ × 𝐻⃗) ⋅ 𝑑𝑠
- The theorem is a generalization of Joule's law and describes the instantaneous power dissipated in a volume
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