Gauss Law for Magnetism *** Biot-Savart Law Ampere Circuital Law Boundary Conditions Potentials Force Inductance Problems Magnetostatics S. R. Zinka [email protected]School of Electronics Engineering Vellore Institute of Technology October 18, 2012 Magnetostatics EE208, School of Electronics Engineering, VIT
41
Embed
Magnetostatics - Arraytool · 2012-10-04 · Gauss Law for Magnetism ***Biot-Savart LawAmpere Circuital LawBoundary ConditionsPotentialsForceInductanceProblems Magnetostatics S. R.
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Gauss Law for Magnetism *** Biot-Savart Law Ampere Circuital Law Boundary Conditions Potentials Force Inductance Problems
Gauss Law for Magnetism *** Biot-Savart Law Ampere Circuital Law Boundary Conditions Potentials Force Inductance Problems
Outline
1 Gauss Law for Magnetism ***
2 Biot-Savart Law
3 Ampere Circuital Law
4 Boundary Conditions
5 Potentials
6 Force
7 Inductance
8 Problems
Magnetostatics EE208, School of Electronics Engineering, VIT
Gauss Law for Magnetism *** Biot-Savart Law Ampere Circuital Law Boundary Conditions Potentials Force Inductance Problems
Outline
1 Gauss Law for Magnetism ***
2 Biot-Savart Law
3 Ampere Circuital Law
4 Boundary Conditions
5 Potentials
6 Force
7 Inductance
8 Problems
Magnetostatics EE208, School of Electronics Engineering, VIT
Gauss Law for Magnetism *** Biot-Savart Law Ampere Circuital Law Boundary Conditions Potentials Force Inductance Problems
Coulomb’s Law for Magnetism - Discrete ***
The force ~Fmir
(ir stands for irrotational) acting on the charge qm located at~r, due to the presence ofthe charge q′m located at ~r′ in an otherwise empty space, is given as
~Fmir(~r) =
qmq′m4πµ0
~r−~r′∥∥∥~r−~r′∥∥∥3 = − qmq′m
4πµ0∇
1∥∥∥~r−~r′∥∥∥ = qm~Hir (1)
where vacuum permeability, µ0 = 4π × 107 Hm−1(SI unit).
Magnetostatics EE208, School of Electronics Engineering, VIT
Gauss Law for Magnetism *** Biot-Savart Law Ampere Circuital Law Boundary Conditions Potentials Force Inductance Problems
Coulomb’s Law for Magnetism - Continuous ***
~dHir
=dq′m
4πµ0
~r−~r′∥∥∥~r−~r′∥∥∥3 =
ρ′m
(~r′)
dv′
4πµ0
~r−~r′∥∥∥~r−~r′∥∥∥3
⇒ ~Hir =1
4πµ0
˚V′
ρ′m
(~r′) ~r−~r′∥∥∥~r−~r′
∥∥∥3 dv′ (2)
= −∇
14πµ0
˚V′
ρ′m
(~r′)
∥∥∥~r−~r′∥∥∥ dv′
(3)
Magnetostatics EE208, School of Electronics Engineering, VIT
Gauss Law for Magnetism *** Biot-Savart Law Ampere Circuital Law Boundary Conditions Potentials Force Inductance Problems
Gauss Law for Magnetism ***
According to Helmholtz’s theorem, a well-behaved vector field is completely known if one knows
its divergence and curl. Taking the divergence of~Bir = µ0~Hir = µ0~Fir
mqm
gives
∇ ·~Bir = −∇ ·
∇ 1
4π
˚V′
ρ′m
(~r′)
∥∥∥~r−~r′∥∥∥ dv′
= − 1
4π
˚V′∇2
1∥∥∥~r−~r′∥∥∥ ρ′m
(~r′)
dv′
=
˚V′
δ(~r−~r′
)ρ′m
(~r′)
dv′
=
˚V′
δ(~r′ −~r
)ρ′m
(~r′)
dv′ = ρ′m (~r) . (4)
Since in reality we do not have magnetic charges, ∇ ·~Bir = 0.
Magnetostatics EE208, School of Electronics Engineering, VIT
Gauss Law for Magnetism *** Biot-Savart Law Ampere Circuital Law Boundary Conditions Potentials Force Inductance Problems
Irrotational Property of Magnetic Field Produced byMagnetic Charges ***
We have already stated in the Coulomb’s law for magnetic charges, that magnetic field produced bymagnetic charges (assuming that they do exist) is always irrotational. Now let’s prove that statementmathematically.
Since ∇× [∇α (~r)] ≡~0 for any R3 scalar field, we immediately find that in magnetostatics
∇× ~Hir = −∇×
∇ 1
4πµ0
˚V′
ρ′m
(~r′)
∥∥∥~r−~r′∥∥∥ dv′
= − 1
4πµ0∇×
∇˚
V′
ρ′m
(~r′)
∥∥∥~r−~r′∥∥∥ dv′
= ~0 (5)
i.e., that ~Hir is an irrotational field.
Magnetostatics EE208, School of Electronics Engineering, VIT
Gauss Law for Magnetism *** Biot-Savart Law Ampere Circuital Law Boundary Conditions Potentials Force Inductance Problems
Magnetostatics EE208, School of Electronics Engineering, VIT
Gauss Law for Magnetism *** Biot-Savart Law Ampere Circuital Law Boundary Conditions Potentials Force Inductance Problems
Outline
1 Gauss Law for Magnetism ***
2 Biot-Savart Law
3 Ampere Circuital Law
4 Boundary Conditions
5 Potentials
6 Force
7 Inductance
8 Problems
Magnetostatics EE208, School of Electronics Engineering, VIT
Gauss Law for Magnetism *** Biot-Savart Law Ampere Circuital Law Boundary Conditions Potentials Force Inductance Problems
Current
Current Densities - Notations:
Ie; JLe~dl; ~JS
e ds; ~Jve dv
or
I~dl; ~Kds; ~Jdv
Current Density - Definition:
Ie =dQdt
=ρL
e4x4t
= ρLe vx. (6)
In general,
~Jve = ρe~v (7)
Magnetostatics EE208, School of Electronics Engineering, VIT
Gauss Law for Magnetism *** Biot-Savart Law Ampere Circuital Law Boundary Conditions Potentials Force Inductance Problems
Continuity Equation
The current through a closed surface is given as
Ie =
‹~JS
e · ~ds =‹
~Jve · ~ds =
˚ (∇ ·~Jv
e
)dv = − dQ
dt= − d (
˝ρedv)
dt.
From the above equation,
∇ ·~Jve = − ∂ρe
∂t(8)
Magnetostatics EE208, School of Electronics Engineering, VIT
Gauss Law for Magnetism *** Biot-Savart Law Ampere Circuital Law Boundary Conditions Potentials Force Inductance Problems
Biot-Savart Law - Statement
Magnetic field can also be produced by electric currents (Actually, this is the only way we can pro-duce magnetic field in practice). According to Biot-Savart law, magnetic field ~dH
r(r stands for rota-
tional) at position~r generated by a steady differential current element I~dl is given as
~dHr=
I4π
~dl×(~r−~r′
)∥∥∥~r−~r′
∥∥∥3 . (9)
Magnetostatics EE208, School of Electronics Engineering, VIT
Gauss Law for Magnetism *** Biot-Savart Law Ampere Circuital Law Boundary Conditions Potentials Force Inductance Problems
Finite Line Current
Magnetic field due to a small differential current ele-ment at ~r′ = z′ z is given as
~dH =I0δ (x′) δ (y′) dx′dy′
4π
dz′ z×(~r−~r′
)∥∥∥~r−~r′
∥∥∥3
=I0δ (x′) δ (y′) dx′dy′
4π
dz′ z× (xx + yy + (z− z′) z)[x2 + y2 + (z− z′)2
] 32
.
Converting the above equation into cylindrical coordi-nate system gives
~dH =I0δ (x′) δ (y′) dx′dy′
4π
dz′ z× (ρρ + (z− z′) z)[ρ2 + (z− z′)2
] 32
=I0δ (x′) δ (y′) dx′dy′
4π
ρdz′ φ[ρ2 + (z− z′)2
] 32
.
Magnetostatics EE208, School of Electronics Engineering, VIT
Gauss Law for Magnetism *** Biot-Savart Law Ampere Circuital Law Boundary Conditions Potentials Force Inductance Problems
Finite Line CurrentTo get the total magnetic filed, let’s evaluate the volume integral given below:
~Htotal =
˚V′
~dH =
˚V′
I0δ (x′) δ (y′)4π
ρ[ρ2 + (z− z′)2
] 32
dx′dy′dz′ φ
=I0
4π
ˆ z′=b
z′=a
ρ[ρ2 + (z− z′)2
] 32
dz′ φ.
The above integral can be calculated as shown below:
ˆ z′=b
z′=a
ρ[ρ2 + (z− z′)2
] 32
dz′ = ρ
ˆ ξ=z−b
ξ=z−a
1
[ρ2 + ξ2]32(−dξ) , where ξ = z− z′
= ρ
ˆ τ=tan−1(
z−aρ
)τ=tan−1
(z−b
ρ
) 1
[ρ2 + ρ2tan2τ]32
(ρsec2τdτ
), where ξ = ρtanτ
=1ρ
ˆ τ=tan−1(
z−aρ
)τ=tan−1
(z−b
ρ
) cos τdτ =1ρ
z− a√(z− a)2 + ρ2
− z− b√(z− b)2 + ρ2
.
Magnetostatics EE208, School of Electronics Engineering, VIT
Gauss Law for Magnetism *** Biot-Savart Law Ampere Circuital Law Boundary Conditions Potentials Force Inductance Problems
Circular Loop Current
Magnetic field due to a small differential current ele-ment at ~r′ = aρ at the origin is given as
~dH =I0δ (ρ′ − a) δ (z′) dρ′dz′
4π
ρdφ′ φ×(~r−~r′
)∥∥∥~r−~r′
∥∥∥3
=I0δ (ρ′ − a) δ (z′) ρdρ′dφ′dz′
4π
aza3 .
Taking volume integral for the above equation gives
~H =1a2
˚V′
I0δ (ρ′ − a) δ (z′)4π
ρdρ′dφ′dz′ z
=I0
4πa2
[ˆρ′
ρδ (ρ′ − a) dρ′
] [ˆ φ=2π
φ=0dφ′
] [ˆ z′=+∞
z′=−∞δ (z′) dz′
]z
=I0
4πa2 × a× 2πz =I0
2az.
Magnetostatics EE208, School of Electronics Engineering, VIT
Gauss Law for Magnetism *** Biot-Savart Law Ampere Circuital Law Boundary Conditions Potentials Force Inductance Problems
Outline
1 Gauss Law for Magnetism ***
2 Biot-Savart Law
3 Ampere Circuital Law
4 Boundary Conditions
5 Potentials
6 Force
7 Inductance
8 Problems
Magnetostatics EE208, School of Electronics Engineering, VIT
Gauss Law for Magnetism *** Biot-Savart Law Ampere Circuital Law Boundary Conditions Potentials Force Inductance Problems
Ampere Circuital Law - Statement
Ampere Circuital Law is given (without derivation) as,
˛~H · ~dl = Ienclosed =
¨ (~JS
e · ~ds)=
¨ (~Jv
e · ~ds)
. (10)
The above equation can be combined with Stokes’ theorem to give
∇× ~H =~Jve . (11)
Magnetostatics EE208, School of Electronics Engineering, VIT
Gauss Law for Magnetism *** Biot-Savart Law Ampere Circuital Law Boundary Conditions Potentials Force Inductance Problems
Outline
1 Gauss Law for Magnetism ***
2 Biot-Savart Law
3 Ampere Circuital Law
4 Boundary Conditions
5 Potentials
6 Force
7 Inductance
8 Problems
Magnetostatics EE208, School of Electronics Engineering, VIT
Gauss Law for Magnetism *** Biot-Savart Law Ampere Circuital Law Boundary Conditions Potentials Force Inductance Problems
Boundary Conditions - Tangential Components
12
34
5
6
7x x x x x + + + + +
Using the Stokes’ Theorem,
(ˆ1+
ˆ3+
ˆ2+
ˆ4
)(~H · ~dl
)=
¨ (∇× ~H
)· ~ds(ˆ
1+
ˆ3
)(~H · ~dl
)=
¨ (~Je · ~ds
), (∵ ∇× ~H =~Je)
⇒(
H1tangential −H2
tangential
)4l = K4l
⇒ H1tangential −H2
tangential = K
⇒B1
tangential
µr1−
B2tangential
µr2= K.
Magnetostatics EE208, School of Electronics Engineering, VIT
Gauss Law for Magnetism *** Biot-Savart Law Ampere Circuital Law Boundary Conditions Potentials Force Inductance Problems
Boundary Conditions - Normal Components
12
34
5
6
7x x x x x + + + + +
Using the divergence theorem (and Gauss law for magnetostatics),
(¨5+
¨6+
¨7
)(~B · ~ds
)= Qm,cylinder
⇒(B2
normal − B1normal
)ds = ρmds
⇒ B2normal − B1
normal= ρm
⇒ µr2H2normal − µr1H1
normal=ρm.
Magnetostatics EE208, School of Electronics Engineering, VIT
Gauss Law for Magnetism *** Biot-Savart Law Ampere Circuital Law Boundary Conditions Potentials Force Inductance Problems
Outline
1 Gauss Law for Magnetism ***
2 Biot-Savart Law
3 Ampere Circuital Law
4 Boundary Conditions
5 Potentials
6 Force
7 Inductance
8 Problems
Magnetostatics EE208, School of Electronics Engineering, VIT
Gauss Law for Magnetism *** Biot-Savart Law Ampere Circuital Law Boundary Conditions Potentials Force Inductance Problems
Magnetic Field - Sources
Magnetic field can be produced either from magnetic charges (ρm) or from electric current (I~dl).Magnetic field produced by ρm is always irrotational, whereas magnetic field produced by I~dl isrotational.
~Htotal = ~Hirrotational + ~Hrotational
Magnetostatics EE208, School of Electronics Engineering, VIT
Gauss Law for Magnetism *** Biot-Savart Law Ampere Circuital Law Boundary Conditions Potentials Force Inductance Problems
Gauss Law - Revisit
Gauss Law:
∇ ·~B = ρm
⇒ ∇ ·(~Birrotational +~Brotational
)= ρm + 0
It can be shown that
∇ ·~Birrotational = ρm, and
∇ ·~Brotational = 0 (12)
Magnetostatics EE208, School of Electronics Engineering, VIT
Gauss Law for Magnetism *** Biot-Savart Law Ampere Circuital Law Boundary Conditions Potentials Force Inductance Problems
Ampere Law - Revisit
Ampere Law:
∇× ~H =~Je
∇×(~Hirrotational + ~Hrotational
)=~0 +~Je
It can be shown that
∇× ~Hrotational =~Je, and
∇× ~Hirrotational =~0 (13)
Magnetostatics EE208, School of Electronics Engineering, VIT
Gauss Law for Magnetism *** Biot-Savart Law Ampere Circuital Law Boundary Conditions Potentials Force Inductance Problems
Scalar and Vector Magnetic Potentials
From (12) , (13), and the above diagram, we can introduce two potentials as shown below:
In the above equations, ~A and Vm are called, vector and scalar magnetic potentials, respectively.
Since we don not have magnetic charges, we will concentrate only on vector magnetic potential.
Magnetostatics EE208, School of Electronics Engineering, VIT
Gauss Law for Magnetism *** Biot-Savart Law Ampere Circuital Law Boundary Conditions Potentials Force Inductance Problems
Vector Magnetic PotentialMathematical expression for vector magnetic potential ~A can be derived from Biot-Savart Law asshown below: (The below derivation is not in your syllabus)
~H =1
4π
˚~Je
(~r′)×
(~r−~r′
)∥∥∥~r−~r′
∥∥∥3 dv′
=−14π
˚~Je
(~r′)×∇
1∥∥∥~r−~r′∥∥∥ dv′
=−14π
˚ 1∥∥∥~r−~r′∥∥∥(∇×~Je
(~r′))−∇×
~Je
(~r′)
∥∥∥~r−~r′∥∥∥ dv′ ,∵ ∇× (wA) = w∇×A−A×∇w
=−14π
˚ ~0−∇× ~Je
(~r′)
∥∥∥~r−~r′∥∥∥ dv′ = ∇×
14π
˚ ~Je
(~r′)
∥∥∥~r−~r′∥∥∥ dv′
. (16)
Comparing (16) and (14), we get (you need to remember the below formula)
~A =µ0
4π
˚ ~Je
(~r′)
∥∥∥~r−~r′∥∥∥ dv′ (17)
Magnetostatics EE208, School of Electronics Engineering, VIT
Gauss Law for Magnetism *** Biot-Savart Law Ampere Circuital Law Boundary Conditions Potentials Force Inductance Problems
Outline
1 Gauss Law for Magnetism ***
2 Biot-Savart Law
3 Ampere Circuital Law
4 Boundary Conditions
5 Potentials
6 Force
7 Inductance
8 Problems
Magnetostatics EE208, School of Electronics Engineering, VIT
Gauss Law for Magnetism *** Biot-Savart Law Ampere Circuital Law Boundary Conditions Potentials Force Inductance Problems
Lorentz and Laplace Forces
Lorentz Force:According to the Lorentz force equation, force on a moving charge is given as,
~F =~Fe +~Fm = Q(~E +~u×~B
)(18)
Laplace Force:Similarly, force acting on a differential current element due to an incident magnetic field ~B is givenas
~Fm = I~dl×~B (19)
Magnetostatics EE208, School of Electronics Engineering, VIT
Gauss Law for Magnetism *** Biot-Savart Law Ampere Circuital Law Boundary Conditions Potentials Force Inductance Problems
Ampere Force Law
Magnetostatics EE208, School of Electronics Engineering, VIT
Gauss Law for Magnetism *** Biot-Savart Law Ampere Circuital Law Boundary Conditions Potentials Force Inductance Problems
Outline
1 Gauss Law for Magnetism ***
2 Biot-Savart Law
3 Ampere Circuital Law
4 Boundary Conditions
5 Potentials
6 Force
7 Inductance
8 Problems
Magnetostatics EE208, School of Electronics Engineering, VIT
Gauss Law for Magnetism *** Biot-Savart Law Ampere Circuital Law Boundary Conditions Potentials Force Inductance Problems
Inductance
Magnetostatics EE208, School of Electronics Engineering, VIT
Gauss Law for Magnetism *** Biot-Savart Law Ampere Circuital Law Boundary Conditions Potentials Force Inductance Problems
Mutual Inductance
Magnetostatics EE208, School of Electronics Engineering, VIT
Gauss Law for Magnetism *** Biot-Savart Law Ampere Circuital Law Boundary Conditions Potentials Force Inductance Problems
Magnetic Energy Density
Magnetostatics EE208, School of Electronics Engineering, VIT
Gauss Law for Magnetism *** Biot-Savart Law Ampere Circuital Law Boundary Conditions Potentials Force Inductance Problems
Inductance - Examples
Magnetostatics EE208, School of Electronics Engineering, VIT
Gauss Law for Magnetism *** Biot-Savart Law Ampere Circuital Law Boundary Conditions Potentials Force Inductance Problems
Outline
1 Gauss Law for Magnetism ***
2 Biot-Savart Law
3 Ampere Circuital Law
4 Boundary Conditions
5 Potentials
6 Force
7 Inductance
8 Problems
Magnetostatics EE208, School of Electronics Engineering, VIT
Gauss Law for Magnetism *** Biot-Savart Law Ampere Circuital Law Boundary Conditions Potentials Force Inductance Problems
Biot - Savart Law
1 If~I = 14z A, then the magnetic field intensities ~H atPCart (2,−4, 4) and PCart
(√20, 0, 4
)are given as ____.
2 Find the magnetic field intensity at the center of a circularconductor carrying a current of 10 A with a radius of 2m withorigin as center.
3 The magnetic field intensity at the center of a square conductorof 5m each side carrying a current of 5 A with origin as centerlying in z = 0 plane is ____.
4 The flux φ crossing the plane surface defined by0.5m ≤ ρ ≤ 2.5m and 0m ≤ z ≤ 2m if~B is given by~B = 2
ρ affi.
5 Given~Je = 103 sin θar A/m2 in spherical coordinates, then thecurrent crossing the spherical shell r = 0.02m is ____.
Magnetostatics EE208, School of Electronics Engineering, VIT
Gauss Law for Magnetism *** Biot-Savart Law Ampere Circuital Law Boundary Conditions Potentials Force Inductance Problems
Ampere Circuital Law
Toroid
Magnetostatics EE208, School of Electronics Engineering, VIT
Gauss Law for Magnetism *** Biot-Savart Law Ampere Circuital Law Boundary Conditions Potentials Force Inductance Problems
Boundary Conditions
1 Plane y = 1 carries current ~K = 50z mA/m. Then find the magnetic field intensities at(0, 0, 0) and (1, 5,−3), respectively. (25x,-25x)
2 The plane z = 0 marks the boundary between free space and a dielectric medium with a
dielectric constant of 40. The~E field next to the interface in free space is~E = 13x + 40y + 50z V/m. Then find the~E field on the other side of the interface.
(13x+40y+5/4z)
Magnetostatics EE208, School of Electronics Engineering, VIT
Gauss Law for Magnetism *** Biot-Savart Law Ampere Circuital Law Boundary Conditions Potentials Force Inductance Problems
Potentials
Magnetostatics EE208, School of Electronics Engineering, VIT
Gauss Law for Magnetism *** Biot-Savart Law Ampere Circuital Law Boundary Conditions Potentials Force Inductance Problems
Lorentz Force
force acceleration ... etc
Magnetostatics EE208, School of Electronics Engineering, VIT
Gauss Law for Magnetism *** Biot-Savart Law Ampere Circuital Law Boundary Conditions Potentials Force Inductance Problems
Faraday’s Law
Magnetostatics EE208, School of Electronics Engineering, VIT