Maxwell’s Equations in Free SpaceIntegral Differential
∮ℓ
❑
𝐸 ∙𝑑 ℓ⃗=− 𝜕𝜕𝑡 (∬𝑆
❑
�⃗� ∙𝑑 �⃗�)∮
ℓ
❑
�⃗� ∙𝑑 ℓ⃗=∬𝑆
❑
�⃗� ∙𝑑 �⃗�+𝜕𝜕𝑡 (∬𝑆
❑
�⃗� ∙𝑑 �⃗�)∯𝑆
❑
�⃗� ∙𝑑 �⃗�=𝑞
∯𝑆
❑
�⃗� ∙𝑑�⃗�=0
𝛻× �⃗�=− 𝜕 �⃗�𝜕𝑡
𝛻× �⃗�= �⃗�+ 𝜕 �⃗�𝜕𝑡
𝛻 ∙ �⃗�=𝜌
𝛻 ∙ �⃗�=0
Maxwell’s Equations in Free SpaceIntegral Differential
∮ℓ
❑
𝐸 ∙𝑑 ℓ⃗=− 𝜕𝜕𝑡 (∬𝑆
❑
�⃗� ∙𝑑 �⃗�)∮
ℓ
❑
�⃗� ∙𝑑 ℓ⃗= 𝜕𝜕𝑡 (∬𝑆
❑
�⃗� ∙𝑑𝑆)∯𝑆
❑
�⃗� ∙𝑑 �⃗�=0
∯𝑆
❑
�⃗� ∙𝑑�⃗�=0
𝛻× �⃗�=− 𝜕 �⃗�𝜕𝑡
𝛻× �⃗�=𝜕 �⃗�𝜕𝑡
𝛻 ∙ �⃗�=0
𝛻 ∙ �⃗�=0
Source free
Maxwell’s Equations in Free Space
�⃗� V/m
�⃗� A/m
�⃗�=𝜀0𝐸 C/
�⃗�=𝜇0 �⃗� Wb/
Electric field Magnetic field
Magnetic flux density
Electric flux density
𝜀0=8.8542 ×10−12 F/𝜇0=4𝜋× 10−7 H/
Permittivity
Permeability
Maxwell’s Equations in Free Space
�⃗� V/m
�⃗� A/m
�⃗�=𝜀0𝐸 C/
�⃗�=𝜇0 �⃗� Wb/ “Physical vector quantities may be divided into two classes, in one of which the quantity is defined with reference to a line, while in the other the quantity is defined with reference to an area.” Thus, “there are certain cases in which a quantity may be measured with reference to a line as well as with reference to an area.” -Maxwell
Line related
Area related
}
}
Gradient, Divergence, Curl, and …𝛻=
𝜕𝜕 𝑥 �̂�𝑥+
𝜕𝜕 𝑦 �̂�𝑦+
𝜕𝜕 𝑧 �̂�𝑧𝑉=𝑉 𝑥 �̂�𝑥+𝑉 𝑦 �̂�𝑦+𝑉 𝑧 �̂�𝑧
𝛻 ∙𝑉=𝜕𝑉 𝑥
𝜕 𝑥 +𝜕𝑉 𝑦
𝜕 𝑦 +𝜕𝑉 𝑧
𝜕 𝑧
𝛻×𝑉=| �̂�𝑥 �̂�𝑦 �̂�𝑧
𝜕𝜕𝑥
𝜕𝜕 𝑦
𝜕𝜕 𝑧
𝑉 𝑥 𝑉 𝑦 𝑉 𝑧|
𝛻𝜑=𝜕𝜑𝜕 𝑥 �̂�𝑥+
𝜕𝜑𝜕 𝑦 �̂�𝑦+
𝜕𝜑𝜕 𝑧 �̂�𝑧
Gradient, Divergence, Curl, and …𝑉=𝑉 𝑥 �̂�𝑥+𝑉 𝑦 �̂�𝑦+𝑉 𝑧 �̂�𝑧
𝛻2=𝛻 ∙𝛻=𝜕2
𝜕 𝑥2 +𝜕2
𝜕 𝑦2 +𝜕2
𝜕 𝑧 2
𝛻2𝑉=�̂�𝑥𝛻2𝑉 𝑥+ �̂�𝑦𝛻2𝑉 𝑦+ �̂�𝑧𝛻2𝑉 𝑧
𝛻×𝛻×𝑉=𝛻 (𝛻 ∙𝑉 )−𝛻2𝑉
𝛻=𝜕𝜕 𝑥 �̂�𝑥+
𝜕𝜕 𝑦 �̂�𝑦+
𝜕𝜕 𝑧 �̂�𝑧
Electromagnetic Waves
𝛻× �⃗�=− 𝜕 �⃗�𝜕𝑡
𝛻× �⃗�=𝜕 �⃗�𝜕𝑡
�⃗�=𝜇0 �⃗�
�⃗�=𝜀0𝐸
𝛻× �⃗�=−𝜇0𝜕 �⃗�𝜕𝑡
𝛻× �⃗�=𝜀0𝜕 �⃗�𝜕𝑡
𝛻×𝛻×𝛻× �⃗�=−𝜇0
𝜕𝜕𝑡 (𝛻× �⃗� )
𝛻×𝛻× �⃗�=𝜀0𝜕𝜕𝑡 (𝛻× �⃗� )
Electromagnetic Waves
𝛻× �⃗�=− 𝜕 �⃗�𝜕𝑡
𝛻× �⃗�=𝜕 �⃗�𝜕𝑡
�⃗�=𝜇0 �⃗�
𝛻× �⃗�=−𝜇0𝜕 �⃗�𝜕𝑡
𝛻× �⃗�=𝜀0𝜕 �⃗�𝜕𝑡
𝛻×𝛻×𝛻× �⃗�=−𝜇0𝜀0
𝜕2𝐸𝜕𝑡 2
𝛻×𝛻× �⃗�=−𝜇0𝜀0𝜕2 �⃗�𝜕𝑡 2�⃗�=𝜀0𝐸
Electromagnetic Waves
𝛻×𝛻× �⃗�=−𝜇0𝜀0𝜕2𝐸𝜕𝑡 2
𝛻×𝛻× �⃗�=−𝜇0𝜀0𝜕2 �⃗�𝜕𝑡 2
𝛻×𝛻× �⃗�=𝛻 (𝛻 ∙𝐸 )−𝛻2 �⃗�=−𝜇0𝜀0𝜕2𝐸𝜕𝑡 2
𝛻 ∙ �⃗�=0 𝜀0𝛻 ∙ �⃗�=0 𝛻 ∙𝐸=0
𝛻2 �⃗�=𝜇0 𝜀0𝜕2 �⃗�𝜕𝑡2
Electromagnetic Waves
𝛻×𝛻× �⃗�=−𝜇0𝜀0𝜕2𝐸𝜕𝑡 2
𝛻×𝛻× �⃗�=−𝜇0𝜀0𝜕2 �⃗�𝜕𝑡 2
𝛻×𝛻× �⃗�=𝛻 (𝛻 ∙ �⃗� )−𝛻2 �⃗�=−𝜇0𝜀0𝜕2𝐻𝜕𝑡2
𝛻 ∙ �⃗�=0 𝜇0𝛻 ∙ �⃗�=0 𝛻 ∙�⃗�=0
𝛻2 �⃗�=𝜇0 𝜀0𝜕2 �⃗�𝜕𝑡2
𝛻2 �⃗�=𝜇0𝜀0𝜕2𝐻𝜕𝑡2
Electromagnetic Waves
𝛻2 �⃗�=𝜇0 𝜀0𝜕2 �⃗�𝜕𝑡2 𝛻2 �⃗�=𝜇0𝜀0
𝜕2𝐻𝜕𝑡2𝑣=
1√𝜇0𝜀0
𝛻2 �⃗�=1𝑣2𝜕2 �⃗�𝜕𝑡 2
𝛻2 �⃗�=1𝑣2
𝜕2 �⃗�𝜕𝑡 2
𝜀0=8.8542 ×10−12F/
𝜇0=4𝜋× 10−7H/
𝑣=1
√𝜇0𝜀0
≈ 2.9979225 × 108 m / s
𝑐≡ 2.99792458 ×108m /s 1983
Electromagnetic Waves𝛻2 �⃗�=
1𝑐2
𝜕2 �⃗�𝜕𝑡2
𝛻2 �⃗�=1𝑐2𝜕2𝐻𝜕𝑡2
𝛻2 �⃗�=�̂�𝑥𝛻2𝐸𝑥+�̂�𝑦𝛻2 𝐸𝑦+ �̂�𝑧𝛻2 𝐸𝑧
𝛻2 𝐸𝑥=1𝑐2
𝜕2 𝐸𝑥
𝜕𝑡 2 𝛻2 𝐸𝑦=1𝑐2
𝜕2 𝐸𝑦
𝜕𝑡 2 𝛻2 𝐸𝑧=1𝑐2
𝜕2 𝐸𝑧
𝜕𝑡 2
𝛻2 𝐻𝑥=1𝑐2
𝜕2 𝐻𝑥
𝜕𝑡 2 𝛻2 𝐻 𝑦=1𝑐2
𝜕2𝐻 𝑦
𝜕𝑡 2 𝛻2 𝐻𝑧=1𝑐2
𝜕2 𝐻𝑧
𝜕𝑡 2
Wave equations