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CHAPTER OUTLINE
30.1 The Biot–Savart Law
30.2 The Magnetic Force Between
Two Parallel Conductors
30.3 Ampère’s Law
30.4 The Magnetic Field of a Solenoid
30.5 Magnetic Flux
30.6 Gauss’s Law in Magnetism
30.7 Displacement Current and the
General Form of Ampère’s Law
30.1 The Biot–Savart Law
Biot and Savart arrived at a mathematical expression that gives the magnetic field at
some point in space in terms of the current that produces the field. That expression
is based on the following experimental observations for the magnetic field dB at a
point P associated with a length element ds of a wire carrying a steady current I :
The total magnetic field B created at some point by a
current of finite size
Magnetic Field on the Axis of a Circular Current Loop
30.2 The Magnetic Force Between Two Parallel
Conductors
parallel conductors carrying currents in the same direction
attract each other, and parallel conductors carrying currents in
opposite directions repel each other.
The force per unit length:
Example 4
Interaction between current-carrying wires
2 parallel wires A & B, carry the same size of
current.
•
•
A
B
(a) Draw a few magnetic field lines due to
the current through wire A.
Example 4
Interaction between current-carrying wires
2 parallel wires A & B, carry the same size of
current.
field at B due to
current through A
•
•
A
B
(b) Mark the direction of this magnetic
field at wire B with an arrow.
Example 4
Interaction between current-carrying wires
2 parallel wires A & B, carry the same size of
current.
magnetic force
field at B due to
current through A
•
•
A
B
(c) Mark the direction of magnetic force actin
on B due to magnetic field of the current
through A.
Example 4
Interaction between current-carrying wires
2 parallel wires A & B, carry the same size of
current.
magnetic force
field at B due to
current through A
•
•
A
B
field at A due to
current through B
(d) Deduce the direction of magnetic force
acting on A due to magnetic field of the
current through B.
Example 4
Interaction between current-carrying wires
2 parallel wires A & B, carry same size of
current.
magnetic force
field at B due to
current through A
•
•
A
B
field at A due to
current through B
(e) Do A & B attract or repel each other?
They attract each other.
The force between two parallel wires is used
to define the ampere as follows:
The value 2 X 10-7 N/m is obtained from Equation 30.12 with
I1 = I2 = 1 A and a = 1 m.
30.3 Ampère’s Law
Example 30.4 The Magnetic Field Created
by a Long Current-Carrying Wire
We need to find I’
A superconducting wire carries a current of 104 A.
Find the magnetic field at a distance of 1.0 m from
the wire. ( µ 0 = 4∏ x 10-7 A-m/T)
1. 2.0 x 10-3 T
2. 2. 8.0 x 10-3 T
3. 3. 1.6 x 10-2 T
4. 4. 3.2 x 10-2 T
1.
A 2.0 m wire segment carrying a current of 0.60 A
oriented parallel to a uniform magnetic field of 0.50
T experiences a force of what magnitude?
1.
2.
3.
4.
A. 6.7 N
B. 0.30 N
C. 0.15 N
D. zero
A proton moving with a speed of 3 x 105 m/s perpendicular to a
uniform magnetic field of 0.20 T will follow which of the paths
described below? (qe = 1.6 x 10-19 C and mp = 1.67 x 10-27 kg)
1.
2.
3.
4.
1. a straight line path
2. a circular path of 1.5 cm radius
3. a circular path of 3.0 cm radius
4. a circular path of 0.75 cm radius
A proton which moves perpendicular to a magnetic field of 1.2
T in a circular path of 0.08 m radius, has what speed? (qp = 1.6
x 10-19 C and mp = 1.67 x 10 -27 kg)
1.
2.
3.
4.
A) 3.4 x 106 m/s
B) 4.6 x 106 m/s
C) 9.6 x 106 m/s
D) 9.2 x 106 m/s
A current carrying wire of length 50 cm is positioned
perpendicular to a uniform magnetic field. If the current is
10.0 A and it is determined that there is a resultant force of 3.0
N on the wire due to the interaction of the current and field,
what is the magnetic field strength?
a) 0.6 T
b) 1.50 T
c) 1.85 x 103 T
d) 6.7 x 103 T
A horizontal wire of length 3.0 m carries a current of 6.0 A and is oriented
so that the current direction is 50° S of W. The horizontal component of
the earth's magnetic field is due north at this point and has a strength of
0.14 x 10-4 T. What is the size of the force on the wire?
a)0.28 x 10-4 N
b)2.5 x 10-4 N
c)1.9 x 10-4 N
d)1.6 x 10-4 N
30.4 The Magnetic Field of a Solenoid
A solenoid is a long wire wound in the form of a helix
A reasonably uniform magnetic field can be produced in the space surrounded
by the turns of wire—which we shall call the interior of the solenoid—when the
solenoid carries a current. When the turns are closely spaced, each can be
approximated as a circular loop, and the net magnetic field is the vector sum of
the fields resulting from all the turns.
An ideal solenoid is approached when the turns are
closely spaced and the length is much greater than the
radius of the turns.
We can use Ampère’s law to obtain a quantitative expression for the
interior magnetic field in an ideal solenoid.
•The contributions from sides 2 and 4 are both zero, again
because B is perpendicular to ds along these paths
•The contribution along side 3 is zero because the magnetic
field lines are perpendicular to the path in this region.
•Side 1 gives a contribution to the integral because along
this path B is uniform and parallel to ds
If N is the number of turns in the length !, the total current
through the rectangle is NI.Therefore, Ampère’s law applied
to this path gives
A solenoid with 500 turns, 0.10 m long, carrying a current of 4.0 A and with
a radius of 10-2 m will have what strength magnetic field at its center?
(magnetic permeability in empty space µ 0 = 4∏ x 10 -7 T-m/A)
A) 31 x 10-4 T
2. B) 62 x 10-4 T
3. C) 125 x 10-4 T
4. D) 250 x 10-4 T
1.
30.5 Magnetic Flux
Consider an element of area dA on an arbitrarily shaped surface, as shown in
Figure . If the magnetic field at this element is B, the magnetic flux through the
element is B. dA, where dA is a vector that is perpendicular to the surface and
has a magnitude equal to the area dA. Therefore, the total magnetic flux ΦB
through the surface is:
ΦB=0
Magnetic Flux
Graphical Interpretation of Magnetic Flux
The magnetic flux is proportional to the number of magnetic flux lines
passing through the area.
Magnetic Flux
A General Expression for Magnetic
Flux
  B A  B(Cos) A
The SI unit of magnetic flux is the weber (Wb), named after the German Physicist
W.E. Weber (1804-1891). 1 Wb = 1 T.m2.
EXAMPLE : Magnetic Flux
A rectangular coil of wire is situated in a constant magnetic field whose
magnitude is 0.50 T. The coil has an area of 2.0 m2 . Determine the magnetic
flux for the three orientations, f = 0°, 60.0°, and 90.0°, shown below.
Example 30.8 Magnetic Flux Through a Rectangular Loop
So
30.6 Gauss’s Law in Magnetism
The magnetic field lines of a bar magnet
form closed loops. Note that the net
magnetic flux through a closed surface
surrounding one of the poles (or any other
closed surface) is zero. (The dashed line
represents the intersection of the surface
with the page.)
The electric field lines surrounding an
electric dipole begin on the positive charge
and terminate on the negative charge. The
electric flux through a closed surface
surrounding one of the charges is not zero.
30.7 Displacement Current and the General
Form of Ampère’s Law
When a conduction current is present, the charge on the positive plate changes
but no conduction current exists in the gap between the plates.
For surface S1
For surface S2
because no conduction current passes
through S2
Thus, we have a contradictory situation that arises from the
discontinuity of the current! Maxwell solved this problem by
postulating an additional term on the right side
of Equation 30.13, which includes a factor called the displacement
current Id , defined as
we can express the general form of Ampère’s law (sometimes called the
Ampère–Maxwell law) as
The electric flux through S2 is simply
Hence, the displacement current through S2 is
Example 30.9 Displacement Current in a Capacitor