Lecture 16
PH 102: Physics II
Electromagnetism
Prof. Bipul Bhuyan & Prof. P.K. Giri
Department of Physics
Indian Institute of Technology Guwahati
1
Magnetostatics
1. The Divergence and Curl of B
2. Application of Ampere’s law
3. Magnetic Vector potential
Straight line currents
field has nonzero curl
B
! µ0 I
B=
2π R
! !
µ0 I
µ0 I
"∫ B ⋅dl = "∫ 2π s dl = 2π s "∫ dl = µ0 I
cylindrical coordinates system (s,Φ,z)
! µ0 I
B=
ϕ̂
2π R
! ! µ0 I
"∫ B ⋅ dl = 2π
!
dl = drr̂ + rdϕϕ̂ + dzẑ
µ0 I
1
"∫ r r dϕ = 2π
2π
∫0
dϕ =µ0 I
3
! ! µ0 I
"∫ B ⋅ dl = 2π
µ0 I
1
"∫ r r dϕ = 2π
2π
∫0
dϕ =µ0 I
loop
ϕ2
φ2
φ1
ϕ1
∫ϕ1 dϕ + ∫ϕ2 dϕ = 0
! !
! !
∴"
∫ B ⋅ dl = µ0 Ienc ; Ienc = ∫ J ⋅ da
! !
!
!
"∫ B ⋅ dl = ∫ (∇ × B) ⋅ da
I enc
!
!
∴∇ × B = µ0 J
! !
! !
= ∫ J ⋅ da = µ0 ∫ J ⋅ da
Ampere’s circuital law
The Divergence and Curl of B
!
B ( x, y, z )
!
J ( x ′, y′, z ′ )
!
R = (x − x ′ )iˆ + ( y − y ′ ) ĵ + (z − z ′ ) k̂
Biot-Savart law
! µ0
B=
4π
dτ ′ = dx ′dy ′dz ′
∇ = iˆ ∂x + ĵ ∂ y + ẑ ∂z ∇ ′ = iˆ ∂ x ′ + ĵ ∂ y ′ + ẑ ∂ z ′
! µ0
! R̂
∇⋅ B =
∇ ⋅( J ×
)dτ ′, dτ ′ →dx ′dy ′dz ′
∫
4π
R2
! R̂
! ! ⎛
R̂
R̂ ⎞
∇ ⋅( J ×
)=
⋅ ∇ × J − J ⋅⎜ ∇ ×
2
2 "#$
2 ⎟⎠
⎝
R
R
"%#R%
$
(
)
0
!
∇⋅B = 0
0
the divergence of the magnetic field is zero
∫
! !
J ( r ′ ) × R̂
R2
dτ ′
Biot-Savart law
! µ0
B=
4π
∫
! !
J ( r ′ ) × R̂
R2
! µ0
! R̂
∇× B=
∇ × (J ×
)dτ ′
∫
2
4π
R
dτ ′
! !
!
!
!
!
!
!
! !
∇ × ( A × C) = (C ⋅∇) A − ( A⋅∇)C + A(∇ ⋅ C) − C(∇ ⋅ A)
!
⎛ ! R̂ ⎞ ⎛ R̂
⎞% !
R̂ ! ⎛
R̂ ⎞ R̂
∇×⎜J ×
=⎜
⋅∇ ⎟ J − ( J ⋅∇ )
+ J ⎜∇⋅ ⎟ −
∇⋅ J
⎟
2
2
2⎠
2
!"
#
⎝
⎠
⎝
R 2 ⎠ ⎝!
R
R
R
R
!
#"#
$
!
#"#
$
( − J ⋅∇′ )
(
0
R̂
R2
4πδ 3 ( R̂)
x − x′) ( y − y′)
z − z′)
(
(
=
iˆ +
ĵ +
k̂
R3
R3
R3
)
!
!
!
f (∇ ⋅ A) = ∇ ⋅( fA) − A⋅(∇f )
!
!
!
∇ ⋅( fA) = f (∇ ⋅ A) + A⋅(∇f )
!
!
x − x′
x − x′ !
x − x′
( J ⋅ ∇ ′ )(
) = ∇ ′ ⋅[
J]− (
) ∇′ ⋅ J
3
3
3 !"#
R
R
R
Steady current
x − x′ !
x − x′ ! !
∫v ∇′ ⋅[ R3 J ]dτ = !∫s R3 J ⋅ da → 0
To where J=0
(
)
0
Ampere’s law in differential form
! µ0 ! !
! !
3 ! !
∇× B=
J ( r ′ )4πδ ( r − r ′ )dτ ′ = µ0 J ( r )
∫
4π
!
! !
∇ × B = µ0 J ( r )
Application of Ampere’s law
GEx.5.7: Find the magnetic field a distance s from a long straight wire, carrying a
steady current I
Amperian loop
! !
"∫ B ⋅ dl = µ0 Ienc = µ0 I
s
I
B
! µ0 I
B=
φ̂
2π r
GEx5.8 Find the magnetic field of an infinite uniform surface current K, flowing over
the xy plane
z
sheet of current
y
}
x
l
Ameperian loop
!
I = I x̂
z
!
Babove = −B ŷ
y
x
!
Bbelow = +B ŷ
!
Babove = −B ŷ
z
l
y2
δh
y1
y
!
Bbelow = +B ŷ
! !
"#$%
∫ B ⋅ dl = µ0 I enc= µ0 Kl
z
2Bl
sheet of current
y
}
x
l
Ameperian loop
⎧ µ0
! ⎪⎪ 2 Kĵ for z < 0
B=⎨
⎪− µ0 Kĵ for z > 0
⎪⎩ 2
GEx5.9 Find the magnetic field of a very long solenoid, consisting of n closely wound
turns per unit length on a cylinder of radius R and carrying a steady current I
I
! !
"∫ B ⋅ dl = Bϕ (2π r) = µ0 Ienc = 0
Bϕ = 0
I
R
Ameperian loop
loop 1
"∫1
! !
B ⋅ dl = [B(a) − B(b)]L = µ0 I enc = 0
B(a) = B(b)
K
b
a
loop 2
"∫ 2
! !
B ⋅ dl = BL = µ0 I enc = µ0 NIL
⎧ µ0 NIẑ inside
B=⎨
outside
⎩ 0
2
1