integral form of maxwell's equations
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6.641, Electromagnetic Fields, Forces, and MotionProf. Markus Zahn
I. Maxwell’s Equations in Integral Form in Free Space
1. Faraday’s Law
d ∫E ds = -
dt ∫ µ0 H idaiC S
of E
C
= 0∫ i
di
dt
2.
0
C S S
d= E
dt+ ε∫ ∫ ∫i i i
Circulation Magnetic Flux
EQS form: E ds (Kirchoff’s Voltage Law, conservative electric
field)
MQS circuit form: v = L (Inductor)
Ampère’s Law (with displacement current)
H ds J da da
µ = 4π×10-7 henries/meter0
[magnetic permeability offree space]
Circulation Conduction Displacementof H Current Current
MQS form: H ds = ∫J dai i∫C S
dvEQS circuit form: i = C (capacitor)
dt
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Lecture 1: Integral Form of Maxwell’s Equations
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3. Gauss’ Law for Electric Field
∫ ε0E da = ρdVi ∫S V
ε0 ≈10-9
≈ 8.854 ×10-12 farads/meter36π 1 c = 3≈ × 108 meters/second (Speed of electromagnetic waves inε µ0 0
free space)
4. Gauss’ Law for Magnetic Field
∫ µ0H da = 0iS
In free space:
magnetic
B = µ H0
magneticflux fielddensity intensity (Teslas) (amperes/meter)
5. Conservation of Charge
Take Ampère’s Law with displacement current and let contour C →0
dlim ∫H ids=0= ∫J ida +
dt S
V
dVρ∫
∫ ε0E idaC 0→
C S
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dJ ida + ρdV =0∫ dt ∫
VS
Total current Total chargeleaving volume inside volumethrough surface
6. Lorentz Force Law
f = q E+ v×µ H)( 0
II. Electric Field from Point Charge
2ε0E ida= ε0E 4πr = qr∫S
E =q
r 24πε0r
2
T sin θ= f =q
c 24πε0r
T cos θ=Mg
2 rtan θ= q
=24πε0r Mg 2l
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3⎡2π ε0r Mg⎤12
q = ⎢ ⎥⎣ l ⎦
III. Faraday Cage
d dJ da= i= - ∫ ρdV = - ( ) = dqi -q∫ dt dt dtS
idt = q∫
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IV. Boundary Conditions
1. Gauss’ Continuity Condition
Courtesy of Krieger Publishing. Used with permission.
ε0E ida= ∫σ sdS ⇒ ε0 (E2n - E1n ) dS= σ dSs∫S S
ε0 (E -E ) = σ ⇒n i⎡ε0 (E - E1 )⎤= σ 2n 1n s ⎣ 2 s⎦
2. Continuity of Tangential E
Courtesy of Krieger Publishing. Used with permission.
E ids= (E - E2t ) dl=0 ⇒E - E2t =01t 1t∫C
n× E1
- E2 ) = 0
Equivalent to = Φ2 along boundary
( Φ1
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3. Normal H
∇ µ H = 0 ⇒ H i da = 0i 0 ∫ µ0
S
µ (H - Hbn ) A = 00 an
H = Hbnan
n i ⎡⎣Ha - H
b⎤ = 0⎦
4. Tangential H
∇ ×H= J ⇒ ∫ H i ds = J i da ∫ C S
H ds - Hatds = Kdsbt
H - Hat = Kbt
n × ⎡⎣H a - Hb ⎤ = K⎦
5. Conservation of Charge Boundary Condition
∂ρ∇ i J+ =0
∂t
dJ da + ρdV = 0i∫ dt
V
∫ S
n i ⎣⎡ J
a- J
b ⎦⎤ +
∂∂ tσ = 0s
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