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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 = y − y

' 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