Lecture 10 Notes, 95.657 Electromagnetic Theory I, Fall 2011
Dr. Christopher S. Baird, UMass Lowell
1. Magnetostatics Introduction
 All of the proceeding concepts have been applied to electrostatics: when static electric charges
create static electric fields.
 We now turn to a special case of electrodynamics known as magnetostatics: when electric charges
move, but move in such a way that they create static magnetic fields.
 The current density
J
is a vector field that describes the flow of charge at every point in space.
 The current density is measured as the amount of positive charge per unit area per unit time, with
the direction of the vector indicating the direction of charge flow.
 When the electric current is confined in a wire, it is useful to integrate the current density over the
crosssectional area of the wire and find the total current
I
, or total electric charge flowing through
the wire per unit time.
 If the total charge
Q
V
inside a volume
V
decreases, this means that some charge is flowing out
passed the surface
S
bounding the volume. Charge is always conserved, it can be neither created nor
destroyed:
−
∂
Q
V
∂
t
=
∮
S
J
⋅
n
da
 Use of the divergence theorem leads to the form:
−
∂
Q
V
∂
t
=
∫
V
∇⋅
J
d
x
 Expand the total charge:
−
∂
∂
t
∫
V
d
x
=
∫
V
∇⋅
J
d
x
 Shrink the volume down so that the integrands must be equal everywhere:
−
∂ρ
∂
t
=∇⋅
J
 This is the continuity equation in differential form.
 Magnetostatics is the special case where we assume no build up or depletion of charge at any
point:
∇⋅
J
=
0
Definition of Magnetostatics (constant flow of charge)

The magnetic induction field
B
(also known as the magnetic flux density) is a vector field created
by electrical currents. Magnetic fields directly produce forces on magnets (or other currents).
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2. The BiotSavart Law
 Using
small
straight wires containing currents and compass magnets, Oersted, Biot, and Savart
experimentally found the following properties:
 The magnetic field is directly proportional to the length
dl
of the small wire.
 The magnetic field is directly proportional to the electrical current
I
in the wire.
 The magnetic field is inversely proportional to the square of the distance
r
from the wire.
 The magnetic field points in the direction normal to the plane in which the wire and observation
point lie.
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 Fall '11
 Staff
 Charge, Electrostatics, Mass, Magnetic Force, Magnetic Field, Ampere, current density

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