Study with the several resources on Docsity
Earn points by helping other students or get them with a premium plan
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
Community
Ask the community for help and clear up your study doubts
Discover the best universities in your country according to Docsity users
Free resources
Download our free guides on studying techniques, anxiety management strategies, and thesis advice from Docsity tutors
A detailed explanation of maxwell's equations, fundamental laws governing electromagnetism. It covers key concepts like divergence, curl, gauss's law, faraday-lenz law, and ampere's law. The document also explores the relationship between electric and magnetic fields, polarization, and displacement current. It is a valuable resource for students studying electromagnetism and related fields.
Typology: Study notes
1 / 7
This page cannot be seen from the preview
Don't miss anything!
Lecture 1 - Maxwell’s Equations
Useful definitions
Divergence:
Δ
∇ ⋅ = Δ →
F dS V
F V
v v 1 r^ r lim 0
the change in F
v in the direction it’s pointing
Gauss’ (or Divergence) Theorem:
v v v v
(finds sources in the volume)
Curl:
Δ
∇ × = Δ →
l
v r v v F d s
F n S
1 ˆ lim ο n ˆ^ unit normal to surface
Stoke’s Theorem:
v v v v v
dl
Review of Maxwell’s Equations
Historical, macroscopic approach (MKS)
Gauss' Law:
∫
0
integrate perpendicular component of E-field over closed surface → gives
total charge within surface.
Note that the units of q are Coulombs (C) and for E are N/C, or Volts/meter
(V/m). ∈ 0 has units of Farad/meter. (∈0=8.85x10 -12^ F/m)
Let ρ tot (r) = total charge density (C/m^3 ),
= ∫ V
q (^) tot tot ( r ) dV
v ρ
So that using the divergence theorem,
∫ ∫ ⇒ = = ∇⋅ V V
tot
tot r dV E r dV
q ( ) / 0 ( ) 0
v v v v ρ ε ε
or:
- Coulomb’s Law or Gauss’s Law
q (^) free
E
r
∫ ∫ ⋅ = ∇⋅ V
E r dS EdV
v (^) v v v v ( )
E, D and P
P is the “polarization” of the medium, induced dipole moment per unit
volume. (Coulomb – m per m 3 or C/m^2 )
(i) P = 0 (no dielectric)
(ii) P ≠ 0 (dielectric)
(- note that P
r points from – to +)
P = induced surface charge density (C/m 2 )
ε 0 E = total surface charge density (C/m (^2) )
D = free surface charge density (C/m 2 )
where D is the electric displacement vector
∴ (^) ∫ D^ ⋅^ dS = qfree
r r , (^) ∫ P^ ⋅^ dS =− qbound
r r
. (Note sign.)
( (^) E P ) (^) free D
v v v v v ∇⋅ε 0 + =ρ ≡∇⋅
Hence: (^) bound P
v v ρ = −∇ ⋅ , and
E
r
E
P
r
E P
v v free v v ∇⋅ = − ∇⋅ 0 0
1
ε ε
ρ
dl
Faraday-Lenz Law
Electromotive force ε induced around a circuit, i.e. the work done to move a
unit charge around the circuit
Φ ; Φ =magneticflux ∂
∂ = − B B t
ε (^) (units are Webers, Wb)
Stokes E d S
B dS t
E dl
S
C S v v v
v v v v
= ∇× ⋅
⋅ ∂
∂ ⋅ =−
∫
∫ ∫
( )
c
t
B E ∂
∂ ⇒∇× =−
v v v , the sign coming from Lenz’s Law.
Units of B are Wb/m 2 or Tesla. (1 Tesla = 10 4 Gauss.)
Gauss’ law for B
No magnetic monopoles: following Gauss’ Law ∇^ ⋅ B =^0
v v
i.e.
∫ ∫ ⋅ = = = ∇⋅ s v
B r ds Bdv
v (^) v v v 0
magneticcharge
μ 0
4 Maxwell equations +
D E P r r r
r r r
= +
= +
0
0
(Henceforth we will use J in place of J (^) free .)
Current Sources:
dt
D H J
v v v v ∂ ∇× = +
dt
M J
B v v v v
v
∂ = + ⎭
⎬
⎫
⎩
⎨
⎧ ∇ × − 0 0
free charge ↑^ ↑^ magnetism bound or polarization
current density current density current density
vacuum displacement current density
(^02)
where
