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Electromagnetic Formula Sheet

Vector-Analysis:
Elements
dl
dS
dV

Rectangular
d𝑥𝑎 𝑥 + d𝑦𝑎 𝑦 + d𝑧𝑎 𝑧
d𝑦d𝑧𝑎 𝑥 + d𝑧d𝑥𝑎 𝑦 + d𝑥d𝑦𝑎 𝑧
d𝑥d𝑦d𝑧

Cylindrical
d𝑟𝑎 𝑟 + 𝑟d∅𝑎∅ + d𝑧𝑎 𝑧
𝑟𝑑∅ 𝑑𝑧𝑎𝑟 + 𝑑𝑟𝑑𝑧𝑎∅ + rd𝑟𝑑∅ 𝑎𝑧
rd𝑟𝑑∅ 𝑑𝑧

Haris H
...
𝐴 =

𝜕𝑉
𝜕𝑉
𝜕𝑉
𝑎 +
𝑎 +
𝑎
𝜕𝑟 𝑟 𝑟𝜕∅ ∅
𝜕𝑧 𝑧

∇
...
𝑑𝑙 = 0 , 𝐸 = 4𝜋𝜖 𝑅2 𝑎 𝑅 , 𝑝 = 𝑞𝑑 ,
4𝜋𝜖 𝑜 𝑅2 𝑅12
𝑜
𝑜
𝑜
1
𝜌
1
𝜌𝐑
1
𝜌𝑠
1
𝜌
𝜌
𝜌
∫ 𝑉 𝑅2 𝑑𝑣 𝑎 𝑅 , 𝐸 = 4𝜋𝜖 ∫ 𝑉 𝑅3 𝑑𝑣 , 𝐸 = 4𝜋𝜖 ∫𝑆 𝑅2 𝑑𝑠 𝑎 𝑅 , 𝐸 = 4𝜋𝜖 ∫𝐿 𝑅2𝑙 𝑑𝑙 𝑎 𝑅 , 𝐸 = 2𝜋𝜖𝑙 𝑟 𝑎 𝑟 , 𝐸 = 2𝜖𝑠 𝑎 𝑛 , 𝐸 = −∇𝑉,
4𝜋𝜖 𝑜
𝑜
𝑜
𝑜
𝑜
𝑜
𝐵
𝑞
𝑊
𝑄𝑑𝑐𝑜𝑠𝜃
𝜖𝑠
, 𝑉 = 𝑄 = − ∫ 𝐴 𝐸
...
𝑑𝑠 = ∫ 𝜌 𝑣 𝑑𝑣, ∇ ∙ 𝐷 = 𝜌 𝑣 , 𝑉 𝑑𝑖𝑝𝑜𝑙𝑒 = 4𝜋𝜖𝑅2 , 𝑄 = 𝐶𝑉 , 𝐶 = 𝑑 , 𝐶 =
4𝜋𝜖 𝑜 𝑅
2𝜋𝜖𝐿
𝜌
2
2
2
𝑏 (𝑓𝑜𝑟 𝑐𝑦𝑙𝑖𝑛𝑑𝑟𝑖𝑐𝑎𝑙 𝑐𝑎𝑝𝑎𝑐𝑖𝑡𝑜𝑟), ∇ 𝑉 = − 𝜖 (𝑝𝑜𝑖𝑠𝑠𝑜𝑛′𝑠 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛)(∇ 𝑖𝑠 𝑙𝑎𝑝𝑙𝑎𝑐𝑖𝑎𝑛 𝑜𝑝𝑒𝑟𝑎𝑡𝑜𝑟), ∇ 𝑉 =
𝑙𝑛( )

𝐹=

𝐸=
𝑉=

𝑎

0(𝐿𝑎𝑝𝑙𝑎𝑐𝑒 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛)(𝑛𝑜 𝑐ℎ𝑎𝑟𝑔𝑒 𝑖𝑠 𝑝𝑟𝑒𝑠𝑒𝑛𝑡 𝜌 = 0)

Magnetostatics:
𝐼=

Δ𝑄
Δt

,𝐼 = ∫ 𝐽 ∙ 𝑑𝑠, 𝐽 = 𝜎𝐸(𝐴/𝑚2 )(𝑂𝐻𝑀′ 𝑠𝑙𝑎𝑤 𝑖𝑛 𝑤𝑎𝑣𝑒𝑓𝑜𝑟𝑚), ∇
...
𝐽 =

𝐽 ∙ 𝑑𝑙 = 0,𝐹 𝑚 = 𝑞(𝐕 × 𝐁),

F = 𝐹 𝑚 + 𝐹 𝐸 = 𝑞(𝐄 + 𝐕 × 𝐁)(𝑙𝑜𝑟𝑒𝑛𝑡𝑧 ′ 𝑠𝑓𝑜𝑟𝑐𝑒 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛), ∇
...
𝑑𝑠 = 0(law of conservation of
magnetic flux), ∮𝐶 𝐵 ∙ 𝑑𝑙 = 𝜇 𝑜 𝐼, 𝐵 = ∇ × 𝐴, ∇ ∙ 𝐴 = 0, ∇2 𝐴 = −𝜇 𝑜 𝐽,𝐴 =
𝜇𝑜𝐼
𝑑𝑙 ′ ×𝑹
∮ ( 𝑅3 ), 𝐵 = 𝜇𝐻, 𝑀 = χ 𝑚 𝐻, 1 + χ 𝑚
4𝜋 𝐶
𝜇𝑜𝐼
𝐵∅ = 2𝜋𝑟 𝑎∅ (𝑙𝑖𝑛𝑒 𝑜𝑓 𝑐𝑢𝑟𝑟𝑒𝑛𝑡) , 𝑎∅ = 𝑎 𝑙 × 𝑎 𝑟 ,

𝐵=

𝜇𝑜𝐼
𝑑𝑙 ′
∮𝐶 𝑅
4𝜋
′

,𝐵 =

𝜇𝑜𝐼
∮ ∇
4𝜋 𝐶

×(

𝑑𝑙 ′
),
𝑅

𝐵=

𝜇𝑜𝐼
𝑑𝑙 ′ ×𝑎
∮𝐶 ( 𝑅 2 𝑅 ) ,
4𝜋

= 𝜇 𝑟 , ∮𝐶 𝐻 ∙ 𝑑𝑙 = 𝐼 (𝐴𝑚𝑝𝑒𝑟𝑒 𝑠𝑐𝑖𝑟𝑐𝑢𝑖𝑡𝑎𝑙 𝑙𝑎𝑤), Φ = ∮𝐶 𝐴 ∙ 𝑑𝑙 = ∮𝑆 𝐵 ∙ 𝑑𝑠 ,
Φ12 = 𝐿12 𝐼1 , Λ12 = 𝑁2 Φ12 = 𝐿12 𝐼1 ,𝐹 𝑚 = 𝐼 ∫ 𝑑𝑙 × 𝐵

Time Varying Fields and Maxwell’s Equations:
𝑉 𝑒𝑚𝑓 = −

𝑑Φ
𝑑𝑡

, 𝑉 𝑒𝑚𝑓 = ∮ 𝐸 ∙ 𝑑𝑙 = − ∫

𝜕𝐵
𝜕𝑡

∙ 𝑑𝑠 , 𝑉 𝑒𝑚𝑓 = − ∮(𝑢 × 𝐵) ∙ 𝑑𝑙 ,

𝐌𝐚𝐱𝐰𝐞𝐥𝐥 𝐞𝐪𝐮𝐚𝐭𝐢𝐨𝐧𝐬
Differential form
𝜕𝐵
∇× 𝐸 =−
𝜕𝑡
𝜕𝐷

∇× 𝐻 = 𝐽+

𝜕𝑡

Integral form
𝑑Φ
∮ 𝐸 ∙ 𝑑𝑙 = −
𝑑𝑡
𝜕𝐷
∮ 𝐻 ∙ 𝑑𝑙 = 𝐼 + ∮
∙ 𝑑𝑠
𝑆 𝜕𝑡

Significance
Faraday’s Law
Ampere’s circuital Law

∇ ∙ 𝐷 = 𝜌𝑣

∮ 𝐷 ∙ 𝑑𝑠 = 𝑄

Gauss’s Law

∇
...
𝑑𝑠 =
𝜖𝑜
𝑆

Magnetostatics
𝐹 𝑚 = 𝑞(𝐕 × 𝐁)

∮ 𝐸
...
𝐵 = 0
∇× 𝐵 = 𝜇𝑜𝐽
∮ 𝐵
...
com



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