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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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