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

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}

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