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Chapter 30
Sources of the Magnetic Field
(Cont.)
PHY 1361
Dr. Jie Zou
1
Outline
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The magnetic force between two
parallel conductors (30.2)
Ampère’s Law (30.3)
The magnetic field of a solenoid (30.4)
Magnetic flux (30.5)
Gauss’s law in magnetism (30.6)
Magnetism in matter (30.8) (very brief
discussion)
PHY 1361
Dr. Jie Zou
2
The magnetic force between
two parallel conductors
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Example (problem #17):
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What can we learn from this example:
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PHY 1361
In the figure shown, the current in the long,
straight wire is I1 = 5.00 A and the wire lies in
the plane of the rectangular loop, which carries
the current I2 = 10.0 A. The dimensions are c =
0.100 m, a = 0.150 m, and ℓ = 0.450 m. Find
the magnitude and direction of the net force
exerted on the loop by the magnetic field
created by the wire.
The force between parallel conductors is
attractive if the currents are parallel and
repulsive if the currents are antiparallel.
Dr. Jie Zou
3
Ampère’s Law
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Ampère’s Law: The line integral of Bds
around any closed path equals 0I, where I
is the total steady current passing through
any surface bounded by the closed path.
 B  ds   0 I
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Example (problem #21)
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Orange: conductors
Green: rubber
PHY 1361
Ampère’s Law is useful for calculating the
magnetic field of current configurations having
a high degree of symmetry.
The figure shown is a cross-sectional view of a
coaxial cable. In a particular application, the
current in the inner conductor is 1.00 A out of
the page and the current in the outer conductor
is 3.00 A into the page. Determine the
magnitude and direction of the magnetic field at
points a and b.
Dr. Jie Zou
4
The magnetic field of a
solenoid
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A solenoid: a long wire wound in the form
of a helix.
An ideal solenoid: when the turns are
closely spaced and the length is much
greater than the radius of the turns.
Properties:
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(a)
(b)
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(a) A tightly wound solenoid of
finite length; (b) Cross-sectional
view of an ideal solenoid.
PHY 1361
The external field is close to zero.
The interior field is uniform and parallel to
the axis: B = 0(N/l)I = 0 nI.
n = N/l: the number of turns per unit length.
Example (problem #29):
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What current is required in the windings of a
long solenoid that has 1000 turns uniformly
distributed over a length of 0.400 m, to
produce at the center of the solenoid a
magnetic field of magnitude 1.00 × 10-4 T?
Dr. Jie Zou
5
Magnetic flux
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General definition:  B   B  dA
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Special case: a plane of area A in a
uniform field B.
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(b)
(a)
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B = BA cos ; : the angle between B
and A.
(a) If B // the plane, =90°, B=0;
(b) If B  the plane, =0°, B=BA (the
maximum).
Example (problem #33):
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PHY 1361
SI unit: weber (Wb); 1 Wb = 1 Tm2.
A cube has edge length ℓ = 2.50 cm. A
uniform magnetic field B = (5i + 4j +
3k) T exists throughout the region. (a)
Calculate the flux through the shaded
face. (b) What is the total flux through
the six faces?
Dr. Jie Zou
6
Gauss’s law in magnetism
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Gauss’s law in magnetism: the
net magnetic flux through any closed
surface is always zero, i.e.  B  dA  0
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Example (problem #35):
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PHY 1361
Magnetic fields are continuous and form
closed loops. Magnetic field lines do not
begin or end at any point.
Electric field lines originate and terminate
on electric charges.
The hemisphere is in a uniform magnetic
field that makes an angle θ with the
vertical. Calculate the magnetic flux
through (a) S1 and (b) S2.
Dr. Jie Zou
7