PHYS 532. L6 1

Chapter 5 Summary5.1 Introduction and Definitions

• Definition of Magnetic Flux Density B– To find the “magnetic flux density” B at x, place a

small magnetic dipole µ at x and measure the torqueon it:

N = µ × B(x) (5.1)

– ρ=charge density (coul/m3)

– J=current density (amps/m2)

– The torque tries to turn the dipole in the direction ofB.

• Conservation of charge condition:∂ρ∂t

+ ∇ ⋅J = 0 (5.2)

PHYS 532. L6 2

5.2 Biot-Savart Law

• Force on a volume V:

F = J × B d3x

V∫

N = x × (J × B) d3x

V∫

• Torque on volume V:

• Magnetic force on a point charge q moving at v:

(5.12)

(5.13)

F = qv × B

PHYS 532. L6 3

• Integral form of Biot-Savart Law:

B(x) =µo

4πJ(x' )×

x − x'

x − x' 3∫ d3x' =µo

4π∇ ×

J(x' )x − x'∫ d3x' (5.14)

(5.16)

• Differential equations:– Equation (5.16) obviously implies

∇ ⋅B = 0

– For static conditions (∇⋅J=0), (5.16) implies

∇ × B = µoJ

(5.17)

(5.22)

– Integrating (5.22) over a closed area gives Ampere’slaw:

B ⋅ dlC∫ = µo I

where I is the current threading the closed loop C.

(5.25)

5.3 Differential Equations ofMagnetostatics and Ampere’s Law

where µo=4π×10-7

PHYS 532. L6 4

5.4 Vector Potential

• Since ∇⋅B=0, any magnetic flux density can berepresented in terms of a vector potential:

B = ∇ × A (5.27)

• Integral expression for A:

A(x) =µo

4πJ(x')x - x'∫ d3x' + ∇Ψ(x) (5.28)

PHYS 532. L6 5

5.6 Magnetic Fields of a LocalizedCurrent Distribution

• In (5.28), use an expansion like the one we used toget the multipole expansion in electrostatics:

1x − x'

=1x

+x ⋅ x'

x3 + ...

• First term gives zero (no magnetic monopole).

• Second term gives

(5.28)

A(x) =µo

4πm × x

x 3 (5.55)

where m is the magnetic dipole moment:

m =12

x'∫ × J(x')d3x' (5.54)

If the current is a plane loop, the magnetic momenthas magnitude equal to the current in the loop timesthe area of the loop.

B(x) =µo

4π3x(x ⋅ m) − m

x 3 +8π3

mδ (3)(x)

Looks like E of electric dipole

Extra term tomake integralsright includingorigin

(5.64)

PHYS 532. L6 6

5.7 Force and Torque on and Energy ofa Localized Current Distribution in an

External Magnetic Induction

• Force on a dipole in an external magnetic inductionB:

F = ∇(m ⋅ B) (5.69)

N = m × B (5.71)

Dipole aligned with B is drawn toward the region ofstrong field.

PHYS 532. L6 7

5.8 Macroscopic Equations, BoundaryConditions on B and H

• For materials, define M = magnetic moment perunit volume– The magnetic effect of that continuum of dipoles is

equivalent to a current distribution (calledmagnetization current)

JM = ∇ × M (5.79)

PHYS 532. L6 8

• Define the “magnetic field” H:

• Magnetic Maxwell equations in a medium:– The flux-conservation Maxwell equation, which is

homogeneous, remains the same as in a vacuum:

∇ ⋅B = 0

– For the Ampere’s Law Maxwell equation, split thecurrent into two parts: magnetization current andcurrent carried by free particles:

∇ × B = µo J + ∇ × M( )

(5.75)

(5.80)

Current carried by free charges

H =1

µoB − M (5.81)

• The magnetic field Maxwell equations in a mediumthen become

∇ ⋅B = 0

∇ × H = J(5.82)

• For a linear, isotropic medium, it is convenient towrite

B = µH

where µ is called the “magnetic permeability.”

(5.84)

• For ferromagnetic materials, B is a nonlinearfunction of H.

PHYS 532. L6 9

• Note that∇ ⋅H = − ∇ ⋅M

• If M is specified (as is typically the case forproblems involving ferromagnetic materials), then

∇ × H = 0

and the equations of magnetostatics are equivalentto the equations of electrostatics.

• Among working scientists (as opposed to textbookauthors), there is often confusion about H and B.Plasma physicists usually use the symbol B andavoid H, but they refer to B as the magnetic field,which is politically incorrect.

• Boundary conditions:– The normal component of B is conserved at the

boundary between magnetic materials.

– The tangential component of H is conserved at sucha boundary if the only current flowing on theboundary is magnetization current.

PHYS 532. L6 10

5.15 Faraday’s Law of Induction• Integral form:

E = −k

dF

dt(5.135)

E = E' ⋅dlC∫ (5.134)

where– the EMF in the circuit is defined by

and E’ is the electric field in the rest frame of thecurve C, which may be moving.

– The magnetic flux threading the circuit C is definedby

F = B ⋅ ˆ n daS∫ (5.133)

where S = surface bounded by C.

– In SI units, the constant k is 1. For gaussian units, itis 1/c.

• Differential equation form of Faraday’s Law (SIunits):

∇ × E +∂B∂t

= 0 (5.134)

• Faraday’s Law was an experimental discovery.However, it could almost have been derived fromconsideration of the properties of E and B underGalilean transformation.

E' = E + v × B

B' = B

(5.142)

PHYS 532. L6 11

• Lenz’s Law:– Intuitive way to see the sign of the induced electric

field:

The induced current (and accompanying magneticflux) is in such a direction as to oppose the change offlux through the circuit:

Suppose B is out of page and increasingwith time. Induced E is clockwise. In aconducting wire, it would drive aclockwise current, which, by Biot-Savartlaw, would cause a magnetic field intothe page.

PHYS 532. L6 12

5.16 Energy in the Magnetic Field

• Magnetic fields by themselves don’t change particleenergies, because the magnetic force isperpendicular to the particle velocity.– Therefore, it is not possible to discuss magnetic

energy in the context of magnetostatics.

– However, because time-dependent magnetic fieldsimply electric fields, creation of a magnetic fieldconfiguration requires energy.

• Expressions for change in energy associated withchange in magnetic field:

δW = δA∫ ⋅ J d3x = δB∫ ⋅ H d3x

– In a linear medium, with B proportional to H,

(5.144, 5.147)

W =12

B∫ ⋅H d3x =12

J∫ ⋅ A d3x (5.148, 5.149)