Download Tuesday, October 23 rd

Today’s lecture
V. Magnetostatics (in vacuum)
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Magnetic field
Lorentz force
Current and current densities
Continuity equation
The Law of Biot-Savart
Magnetic field B and Lorentz Force
In electrostatics, we studied forces and fields of stationary
charges.

• Stationary charge is surrounded by electric field E
What happens if the charges are moving?
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• 

Moving charge is surrounded
by E

and magnetic field B
The Lorentz Force describes the force on a charged
particle Q in an electric and magnetic field.
It has been found experimentally.

  
F = Q(E + v × B )

Felec

Fmag
The force acting on the component v⊥
perpendicular to the magnetic field is
a central force and forces the charge on a circle.
The component v|| parallel to the field is
not affected.⇒ Trajectory becomes a helix.
No work done by magnetic force
For a charge Q moved along a displacement
the work done is:
dWmag
 
dl = v ⋅ dt


  
= Fmag ⋅ dl = Q ⋅ (v × B )⋅ v ⋅ dt = 0
0
Magnetic forces do no work!
Magnetic forces can only change the direction (of the
Velocity) of charged particles. They cannot change the
magnitude (of their velocity).
Example: J.J. Thomson’s experiment (1897) for q/m
1. Determine v by measuring E/B:
2. Determine q/m by measuring R with E turned off.
Electric currents - line charge
Current is charge per unit time passing a given point.
Convention: positive current points in the direction
the positive charges are flowing.
Currents only relate to the moving charges.
It is measured in Ampere: 1 A=1 C/s.
A line charge traveling in a wire with speed v constitutes a
current of a line charge:


I = λ ⋅v
The force on a wire with current I in a magnetic field B is:

 
Fmag = ∫ (v × B )⋅ dq
 
= ∫ (v × B )⋅ λ ⋅ dl
 
= ∫ (I × B )⋅ dl
 
= ∫ I dl × B
 
= I ∫ dl × B “magnetostatic case” I const in wire
(
(
)
)
Surface and volume current densities
Charge flowing over a surface is described by the
surface current density:

 dI

K :=
=σ ⋅v
dl⊥
σ is the mobile surface charge density. K is current per
unit width perpendicular to the flow. K will change in
general over the surface due to changes in σ or v.
The magnetic force on the surface current in a magnetic
field is 

Fmag

= ∫∫ (v × B )⋅ σ ⋅ da
 
= ∫∫ (K × B )⋅ da
The flow of charge through a volume is described by
means of the volume current density:

 dI

J :=
= ρ ⋅v
da⊥
ρ is the mobile volume charge density.
The magnetic force on the volume current in a magnetic



field is
F =
v × B ⋅ ρ ⋅ dτ
mag
∫∫∫ (
)
 
= ∫∫∫ (J × B )⋅ dτ
Continuity equation
Current crossing a surface S
 
I = ∫∫ J ⋅ da⊥ = ∫∫ J ⋅ da
S
S
The total charge per unit time leaving a volume V is
 
 
∫∫ J ⋅ da = ∫∫∫ ∇ ⋅ J ⋅ dτ
S
V
( )
Due to the conservation of charge
 
d
δρ
∫∫∫ ∇ ⋅ J ⋅ dτ = − dt ∫∫∫ ρ ⋅ dτ = − ∫∫∫ δ t ⋅ dτ
V
V
V
( )
This holds for any volume V and therefore the result is the
mathematical expression for local charge conservation,
the Continuity equation:
  δρ
∇⋅ J = −
δt
Steady currents and the law of Biot-Savart
Stationary charges ⇒constant electric fields ⇒electrostatics
Steady currents⇒constant magnetic fields⇒magnetostatics
A steady current means that charge is continuously flowing
And not piling up anywhere, which means:
δρ
=0
δt
In magnetostatics the continuity equation becomes
 
∇⋅ J = 0
The magnetic field of a steady line current is given by the
Law of Biot-Savart:
 
 
  µ0 I × ur −r '
µ 0 dl '×ur − r '
B(r ) =
I∫   2
  2 dl ' =
∫
4π (r − r ')
4π
(r − r ')
• Steady current (I=const) produces a
time-independent magnetic field B,
• Integration along flow of the current,
• Superposition principle applies,
• µ0 =4π×10-7 N/A2 is permeability of free space
Biot-Savart Law for cont. current distributions
The Biot-Savart Law for line currents:
 
  µ0 I (r ' ) × ur −r '
B(r ) =
  2 dl '
∫
4π
(r − r ')
The Biot-Savart Law for surface currents:
 

 
µ0 K (r ' ) × ur − r '
B(r ) =
  2 da '
∫∫
4π S (r − r ')
The Biot-Savart Law for volume currents:
 

  µ0
J (r ' ) × ur −r '
B(r ) =
  2 dτ '
∫∫∫
4π V (r − r ')
Note: Due to our definition of
source point and field point,
the integral is over primed
Coordinates. Div and curl
Are taken over
unprimed coordinates!
field point

r = (x, y , z )
  
d = r − r'
dτ '
source
point
Examples
Example: (5.9 a and b)
Find the magnetic field at point P for each of the steady
current configurations.