page 1
PHY2323
Formula Sheet
Maxwell’s Equations
⃗ = 𝝆𝒗
𝛁 ∙ ⃗𝑫
⃗ =𝟎
𝛁 ∙ ⃗𝑩
⃗⃗
𝝏𝑩
𝛁 × ⃗𝑬 = −
𝝏𝒕
⃗⃗
𝝏𝑫
⃗⃗ = 𝑱 +
𝛁 × ⃗𝑯
𝝏𝒕
Electrostatics
Coulomb’s law
𝐹⃑ = 𝑞𝐸⃗⃑
Electric field
Potential
1
𝜌𝑙 (𝑟⃑′)
(𝑟⃑ − 𝑟⃑ ′ )𝑑𝑙′
𝐸⃗⃑ (𝑟⃑) =
∫
4𝜋𝜖0 𝑙 |𝑟⃑ − 𝑟⃑ ′ |3
1
𝜌𝑠 (𝑟⃑′)
(𝑟⃑ − 𝑟⃑ ′ )𝑑𝑠 ′
𝐸⃗⃑ (𝑟⃑) =
∫
4𝜋𝜖0 𝑠 |𝑟⃑ − 𝑟⃑ ′ |3
1
𝜌𝑣 (𝑟⃑′)
(𝑟⃑ − 𝑟⃑ ′ )𝑑𝑣′
𝐸⃗⃑ (𝑟⃑) =
∫
4𝜋𝜖0 𝑣 |𝑟⃑ − 𝑟⃑ ′ |3
𝑄 = ∫𝑙 𝜌𝑙 (𝑟⃑ ′ )𝑑𝑙′
𝑄 = ∫𝑠 𝜌𝑠 (𝑟⃑ ′ )𝑑𝑠 ′
1
∫
4𝜋𝜖0 𝑙
1
𝑉(𝑟⃑) =
∫
4𝜋𝜖0 𝑠
1
𝑉(𝑟⃑) =
∫
4𝜋𝜖0 𝑣
𝑉(𝑟⃑) =
𝜌𝑙 (𝑟⃑ ′ )
𝑑𝑙′
|𝑟⃑ − 𝑟⃑ ′ |
′
𝜌𝑠 (𝑟⃑ )
𝑑𝑠′
|𝑟⃑ − 𝑟⃑ ′ |
𝜌𝑣 (𝑟⃑ ′ )
𝑑𝑣′
|𝑟⃑ − 𝑟⃑ ′ |
𝑄 = ∫𝑣 𝜌𝑣 (𝑟⃑ ′ )𝑑𝑣′
𝐵
𝑉𝐵 − 𝑉𝐴 = ∆𝑉 = − ∫𝐴 𝐸⃗⃑ ∙ 𝑑𝑙⃑
𝐸⃗⃑ = −∇𝑉
∮𝑐 𝐸⃗ ∙ 𝑑𝑙 = 0
∇ × 𝐸⃗ = 0 (static)
𝑊 = 𝑞∆𝑉 (Work done to move q through ∆𝑉)
Gauss’s Law and Electric Flux Density
⃗⃑ ∙ 𝑑𝑠⃑ = 𝑄
⃗⃑ = 𝜖𝐸⃗⃑ = 𝜖𝑜 (1 + 𝜒)𝐸⃗⃑ = 𝜖𝑜 𝜖𝑟 𝐸⃗⃑
𝐷
∮𝑠 𝐷
⃗⃑ = 𝜌𝑣
∇∙𝐷
Electric Dipole
𝑝⃑ = 𝑞𝑑𝒛̂
⃗ ∙ 𝑑𝑠
Ψ = ∫𝑆 𝐷
𝑉(𝑟⃑) =
𝑞
𝑑 cos 𝜃
4𝜋𝜖0
𝑟2
Boundary Conditions
Conductors
𝜌
𝐸𝑛 = 𝑠
𝐸𝑡 = 0
𝐸⃗ = 0
𝜖𝑜
=
𝑝⃑∙𝒓̂
4𝜋𝜖|𝑟|2
Dielectrics
𝐸1𝑡 = 𝐸2𝑡
𝜖1 𝐸1𝑛 = 𝜖2 𝐸2𝑛 for ρs = 0
⃗1−𝐷
⃗ 2 ) ∙ 𝑎̂𝑛 = 𝜌𝑠 for ρs ≠ 0
(𝐷
𝜌𝑣 = 0
Dielectrics
⃗ = 𝜖0 𝐸⃗ + 𝑃⃗ = 𝜖𝑜 𝜖𝑟 𝐸⃗ = 𝜖𝐸⃗
𝐷
𝑃⃗⃑ = 𝜖𝑜 𝜒𝐸⃗⃑
Capacitors
𝑄
𝐶=
𝐶=
∆𝑉
Capacitors in parallel 𝐶𝑡𝑜𝑡 = 𝐶1 + 𝐶2 + ⋯
1
1
1
Capacitors in series
= + +⋯
𝐶𝑡𝑜𝑡
𝐶1
𝐴𝜖
𝑑
(parallel plate capacitor)
𝐶2
Energy stored in electric fields:
1
1
⃗ ∙ 𝐸⃗ 𝑑𝑣
𝑊𝐸 = ∫𝑣 𝜌𝑣 (𝑟⃑)𝑉(𝑟⃑)𝑑𝑣 = ∫𝑣 𝐷
2
2
Constants
𝜖0 = 8.85 × 10−12 𝐹/𝑚
𝜇𝑜 = 4𝜋 × 10−7 𝐻/𝑚
1
1
1 𝑄2
2
2
2 𝐶
𝑊𝐸 = 𝑄𝑉 = 𝐶𝑉 2 =
𝑞𝑒 = |𝑒| = 1.6 × 10−19 𝐶
𝑐 = 3 × 108 𝑚/𝑠
Coordinate transformations:
𝐴𝜌
𝑐𝑜𝑠𝜙 sinϕ 0 𝐴𝑥
[𝐴𝜙 ] = [−sinϕ cosϕ 0] [𝐴𝑦 ]
0
0
1 𝐴𝑧
𝐴𝑧
(for capacitors)
𝑚𝑒 = 9.11 × 10−31 𝑘𝑔
𝐴𝑟
𝑠𝑖𝑛𝜃𝑐𝑜𝑠𝜙
[ 𝐴𝜃 ] = [cosθcosϕ
𝐴𝜙
−sinϕ
1
sinθsinϕ cosθ 𝐴𝑥
cosθsinϕ −sinθ] [𝐴𝑦 ]
cosϕ
0
𝐴𝑧
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PHY2323
Electric Current
𝐽 = 𝜌𝑣 𝑣 = 𝜎𝐸⃗ =
1
𝐸⃗
𝜌
𝐿
𝑉 −∫ 𝐸⃗ ∙ 𝑑𝑙
𝑑𝑙
= =
=∫
∑𝑖 𝜎𝑖 𝐴𝑖
𝜎𝐴 𝐼 ∫ 𝜎𝐸⃗ ∙ 𝑑𝑆
𝜕𝜌𝑣
∇∙𝐽 =−
𝜕𝑡
Boundary Conditions
𝑅=
𝐼 = ∫ 𝐽𝑣 ∙ 𝑑𝑆 = ∫ 𝐽𝑠 ∙ 𝑑𝑙
Joule Power
𝑃 = ∫ 𝐸⃗ ∙ 𝐽𝑑𝑣 = ∫ 𝜎𝐸 2 𝑑𝑣 = 𝑉𝐼
𝐽𝑛 1 = 𝐽𝑛 2 ,
𝐽𝑡 1
𝜎1
=
𝐽𝑡 2
𝜎2
Magnetostatics
Biot-Savart
⃗ = 𝜇𝑜 ∫
𝐵
4𝜋
𝐼𝑑𝑙 ×(𝑟 −𝑟 ′)
|𝑟 −𝑟 ′|3
⃗ = 𝜇𝑜 𝑛𝐼 (𝑖𝑛𝑓𝑖𝑛𝑖𝑡𝑒 𝑠𝑜𝑙𝑒𝑛𝑜𝑖𝑑)
𝐵
, 𝐼𝑑𝑙 ↔ 𝐽𝑣 𝑑𝑣 ↔ 𝐽𝑠 𝑑𝑆
Lorentz Force
⃗ , 𝐹𝑚 = ∫ 𝐼𝑑𝑙 × 𝐵
⃗
𝐹𝑚 = 𝑞𝑣 × 𝐵
𝑐
Gauss’ Law for Magnetic Fields
⃗ ∙ 𝑑𝑆 = 0 , ∇ ∙ 𝐵
⃗ =0
∮𝐵
Ampère’s Law
⃗ ∙ 𝑑𝑙 = 𝐼𝑒𝑛𝑐 , ∇ × 𝐻
⃗ = 𝐽𝑣
∮ 𝐻
Torque and Magnetic Dipoles
⃗ , 𝑚
𝜏 =𝑟×𝐹 , 𝜏 =𝑚
⃗⃗ × 𝐵
⃗⃗ = 𝐼∫ 𝑑𝑆 = 𝐼𝐴𝑎̂𝑛
Magnetic Flux
⃗ ∙ 𝑑𝑆
Φ = ∫𝐵
Boundary Conditions
𝐵𝑛 1 = 𝐵𝑛 2 , 𝐻𝑡 1 − 𝐻𝑡 2 = 𝐽𝑠
EMF and the Flux Law
⃗ ) ∙ 𝑑𝑙 , 𝜀 = − 𝑑Φ
𝜀 = ∫ (𝑣 × 𝐵
Faraday’s Law
⃗
𝑑Φ
𝜕𝐵
, ∇ × 𝐸⃗ = −
∮𝑐 𝐸⃗ ∙ 𝑑𝑙 = −
Reluctance
𝑐
𝑑𝑡
Magnetization
⃗ = 𝜇𝑜 (𝐻
⃗ +𝑀
⃗⃗ ) = 𝜇𝐻
⃗ , 𝑀
⃗⃗ = 𝜒𝑚 𝐻
⃗ , 𝜇 = 𝜇𝑜 (1 + 𝜒𝑚 )
𝐵
𝑑𝑡
ℜ = 𝐿/𝜇𝐴
𝜕𝑡
2
page 3
PHY2323
Gradient
(𝑟𝑒𝑐𝑡𝑎𝑛𝑔𝑢𝑙𝑎𝑟 𝑐𝑜𝑜𝑟𝑑. )
(𝑐𝑦𝑙𝑖𝑛𝑑𝑟𝑖𝑐𝑎𝑙 𝑐𝑜𝑜𝑟𝑑. )
(𝑠𝑝ℎ𝑒𝑟𝑖𝑐𝑎𝑙 𝑐𝑜𝑜𝑟𝑑. )
𝜕𝑉
𝜕𝑉
𝜕𝑉
𝑥̂ +
𝑦̂ +
𝑧̂
𝜕𝑥
𝜕𝑦
𝜕𝑧
𝜕𝑉
1 𝜕𝑉
𝜕𝑉
∇𝑉 =
𝜌̂ +
𝜙̂ +
𝑧̂
𝜕𝜌
𝜌 𝜕𝜙
𝜕𝑧
𝜕𝑉
1 𝜕𝑉
1 𝜕𝑉
∇𝑉 =
𝑟̂ +
𝜃̂ +
𝜙̂
𝜕𝑟
𝑟 𝜕𝜃
𝑟𝑠𝑖𝑛𝜃 𝜕𝜙
∇𝑉 =
Divergence
(rectangular coord. )
(cylindrical coord. )
(spherical coord. )
∂Ax ∂Ay ∂Az
+
+
∂x
∂y
∂z
1 ∂
1 ∂
∂
⃗ =
∇∙A
(ρAρ ) +
(Aϕ ) + (AZ )
ρ ∂ρ
ρ ∂ϕ
∂z
1 ∂ 2
1 ∂
1 ∂
⃗ =
(r
)
(sinθA
)
∇∙A
A
+
+
(A )
r
θ
r 2 ∂r
r sinθ ∂θ
r sinθ ∂ϕ ϕ
⃗ =
∇∙A
Laplacian
(𝑟𝑒𝑐𝑡𝑎𝑛𝑔𝑢𝑙𝑎𝑟 𝑐𝑜𝑜𝑟𝑑. )
(𝑐𝑦𝑙𝑖𝑛𝑑𝑟𝑖𝑐𝑎𝑙 𝑐𝑜𝑜𝑟𝑑. )
(𝑠𝑝ℎ𝑒𝑟𝑖𝑐𝑎𝑙 𝑐𝑜𝑜𝑟𝑑. )
𝜕 2𝑉 𝜕 2𝑉 𝜕 2𝑉
∇ 𝑉= 2+ 2+ 2
𝜕𝑥
𝜕𝑦
𝜕𝑧
1 𝜕
𝜕𝑉
1 𝜕 2𝑉 𝜕 2𝑉
∇2 𝑉 =
(𝜌
)+ 2
+
𝜌 𝜕𝜌 𝜕𝜌
𝜌 𝜕𝜙 2 𝜕𝑧 2
1 𝜕 2 𝜕𝑉
1
𝜕
𝜕𝑉
1
𝜕 2𝑉
2
∇ 𝑉= 2
(𝑟
)+ 2
(𝑠𝑖𝑛𝜃 ) + 2 2
𝑟 𝜕𝑟
𝜕𝑟
𝑟 𝑠𝑖𝑛𝜃 𝜕𝜃
𝜕𝜃
𝑟 𝑠𝑖𝑛 𝜃 𝜕𝜙 2
2
3
page 4
PHY2323
Curl:
Trigonometric relations:
sin( 2 )  2 sin( ) cos( )
cos(a)  cos(b)  2 cos(
4
ab
a b
) cos(
)
2
2
page 5
PHY2323
5
page 6
PHY2323
Cartesian coordinate system:
Cylindrical coordinate System:
Spherical coordinate System:
6
page 7
PHY2323
Time Varying Fields:
Motional Electromotive force:
Induced emf:
Magnetic flux linkage:
Self-Inductance:
Linear Magnetic materials:
Induced emf:
Relactance and mmf:
Mutual Inductance:
7
page 8
PHY2323
Coupled Coils in series:
Coupled Coils in parallel:
Total energy in a coil:
For linear magnetic materials:
Magnetic Energy Density:
8