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Maxwell's Equations: Electric Fields and Charges - Prof. Gregory M. Wilkins, Study notes of Electrical and Electronics Engineering

An overview of maxwell's equations, specifically focusing on the electric field and its relationship to charges. Topics include coulomb's law, the electric force between charges, and the calculation of electric fields for various charge distributions. The document also covers the concept of electric potential and its relation to electric fields.

Typology: Study notes

Pre 2010

Uploaded on 08/07/2009

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bg1
t
B
E=×
t
D
JH+=×
v
ρ
=
D
0
=
B
Maxwell’s Equations
point form
What is E?
Coulomb’s Law
Chapter 2 Electric Field
Intensity
pf3
pf4
pf5
pf8
pf9
pfa
pfd
pfe
pff

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∂t

B ∇× E =−

∂ t

∇× H = J + ^ D

∇ ⋅ D = ρ v

∇ ⋅ B = 0

Maxwell’s Equations point form

What is E? Coulomb’s Law

Chapter 2 Electric Field

Intensity

Coulomb's Law

εο=8.854 x 10-12^ F/m permitivity of free space

k =

1 4 πεο

F

Q Q 4 r

1 2 o 2

= πε

F k Q Q e r 1 2 = (^2) Newtons

Electric Field

F t a 1

1 t o 1t

Q Q

4 R

πε t

F

t a 1 E Q

Q

t^4 R

1 o 1t

πε t N C

V = (^) m 1 volt = (^) CJ = NmC ⇒ (^) mV= NC In general:

E = (^4) πε QoR 2 aR

!! aR is the unit vector from the charge to the point where the field is to be measured

Q 1

R 1t Qt

r t

r 1

a 1t F t

R 1t = r t - r 1

Electric Field

Q

Qt R = r - r`

r

r` E

Origin

(x,y,z`)

(x,y,z)

Imaginary Unit Test Charge

E = aR

Q

4 πε oR 2

[ ]

forn charges 4

E(r)

4 '

' '

' 4 '

E(r)

1 0

0 2 3

m

n m m

Qm

o

Q Q

a r r

r r

r r r r

r r r r

∑ − −

=

= − −

− −

=

πε

πε πε

0 3 /^2

0

( ')^2 ( ')^2 ( ')^2

[( ') ( ') ( ') ] 4

4 2

^ − + − + − 

= − + − + −

=

x x y y z z

Q x x y y z z

'

' '

Q E(r)

a x a y a z

r r

r r r r

πε

πε

For charge elsewhere:

3 0

3 0

3 0

3

r r

r r

r r

r r

r r

r r

r r

r r

πε

ρ

πε

ρ

πε

ρ

πε

dv

ds

dl

Q

v

s

l

E

Line C/m

Sheet C/m^2

Volume C/m^3

Charge Distributions

Line charge

2 2 23 2 2 2 2

2 2 23 2 2

2 2 2 2 1 2 2 L 2 3

(x a ) (z-z'^ )

dx

1 (x a )

dx

( (z-z') )

(z-z') (a )

dx

=

=

=−

= −

ρ

ρ ρ

ρ ρ

ρ ρ

ρ ρ

L L L

L

x

x

x a

x

x

dx (a a

x (^2) (a (^22)

=

x^2 x

3 2 2

1 2

1 ) )

and a ρ

dx dz'

1 dz'

dx

let x (z -z')

=

= −

= −

=

−∞

−∞

Z E L^ a a o (^2222) (z-z')^2

(z-z')

(z -z') 4 ρ ρ ρ

ρ π ε

ρ ρ

Finally:

( )

ρ

ρ

π ε ρ

ρ

π ε ρ

ρ

E a

E a a

o

o

2

0

2 4

L

Z

L

=

  + 

  

y z

x y z r a a

r a a a ' y ' z '

x y z = +

= + +

Electric Field Due to a Sheet Charge

'^2 ( ')^2 ( ')^2

' ( ') ( ') x y y z z

x y y z z − = + − + −

− = + − + − r r

r r ax ay az

Q = (^) ∫ ρ s d s d s= d y' d z'

( ' ) 4 [( ') ]

y' z'

4 [ ( ') ( ') ]

y' z'[ ( ') ( ') ]

2 2 2 2 2 2 3

2 2 2 2 3

∫ ∫

∫ ∫ ∞ −∞

∞ −∞

∞ −∞

∞ −∞

= + − − +

=

  • − + −

= + − + −

c x z z y y c

sxd d

x y y z z

sd d x y y z z

o

o

πε

ρ

πε

E ρ^ ax^ ay az

r`

d s=dy’dz’

(y,z)

(x,y,z)

ρs

2 2 23 2 u^2 c^2

1 u (u c )

u

=

∫ (^) c

d du dy '

( ') = −

u = yy

' a x ( ') ( ')

' 4 ( ')

1 (^2 2) y y (^2) x (^2) z z 2 dz

y y x z z

sx o

−∞

− ∞ 

 

 

 

 − + + −

− −

= (^) ∫ ρ πε

− ∞ + −

= 2 2 ax ( ')

' ( 2 ) 4

1 x z z

sxdz o

ρ πε

∫ (^) + = a

a

v

v a

dv

arctan 2 2 a x

a x

dv dz

v z z

=

=

= −

= −

2 2

'

( ' )

n o

s a

E =

Multiple Sources

ET=E 1 +E 2 +E 3 +…En