Transcript Chapter 30 Sources of the Magnetic Field Dr. Jie Zou PHY 1361

Chapter 30
Sources of the Magnetic Field
Dr. Jie Zou
PHY 1361
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Introduction
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The first evidence of the close connection
between electricity and magnetism was
obtained accidentally by the Danish
scientist Hans Christian Oersted in 1820.
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Hans Christian Oersted
(1777-1851)
Dr. Jie Zou
An electric current produces a magnetic field.
Biot and Savart performed quantitative
experiments on the force exerted by an
electric current on a nearby magnet.
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Biot-Savart law: a mathematical expression to
calculate the magnetic field produced at some
point in space by a small current element.
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Outline
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The Biot-Savart Law (30.1)
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Mathematical expression
Applications in finding the total magnetic
field produced by various current
distributions
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Dr. Jie Zou
Example 1: Thin, straight current-carrying wire
Example 2: Circular current loop
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The Biot-Savart Law
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The magnetic field dB at a point P
associated with a length element ds of
a wire carrying a steady current I is
given by:
0 Ids  rˆ
dB 
4 r 2
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Dr. Jie Zou
0 = 4  10-7 Tm/A, the permeability of
free space.
(1) The direction of dB is perpendicular to
both ds and r̂ , and thus perpendicular the
plane formed by ds and r̂ .
(2) The magnitude of dB is proportional to
I, ds, and sin , but inversely proportional
to r2.
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Quick Quiz: where is the
magnetic field the greatest?
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Consider the current in the length of wire
shown in the figure below. Rank the points A,
B, and C, in terms of magnitude of the
magnetic field due to the current in the
length element shown, from greatest to least.
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Total magnetic field due to a
current distribution
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Evaluate B by integration:
0 I ds  rˆ
B
4  r 2
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(1) The above equation follows the
principle of superposition.
(2) The integral is taken over the entire
current distribution.
(3) The integrand is a cross product and
therefore a vector quantity.
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Example 1: Thin, straight currentcarrying wire
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Consider a thin, straight wire carrying a
constant current I and placed along the x
axis as shown in the figure below.
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Dr. Jie Zou
(1) Determine the magnitude and direction of
the magnetic field at point P due to this
current.
Answer: B 
0 I
cos 1  cos  2 , out of the page.
4a
(2) Find the magnetic field at P in the limit of
an infinite long, straight wire.
Answer: B = 0I/(2a), out of the page.
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Magnetic field surrounding a long,
straight current-carrying wire
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Direction of B:
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Magnitude of B: B = 0I/(2a)
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Dr. Jie Zou
The magnetic field lines are circles
concentric with the wire and lie in planes
perpendicular to the wire.
The right-hand rule: grasp the wire with
the right hand, positioning the thumb
along the direction of the current. The
four fingers wrap in the direction of the
magnetic field.
B is constant on any circle of radius a.
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Example 2: Curved wire
segment
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Calculate the magnetic field at point O
for the current-carrying wire segment
shown. The wire consists of two straight
portions and a circular arc of radius R,
which subtends an angle . The
arrowheads on the wire indicate the
direction of the current.
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Dr. Jie Zou
Answer: B = 0I/(4R), into the page at O.
Can you also find the magnetic field at the
center of a circular wire loop of radius R that
carries a current I?
Answer: B = 0I/(2R),
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Magnetic field lines
surrounding a current loop
(a-b) Magnetic field lines surrounding a current loop. (c)
Magnetic field lines surrounding a bar magnet. Note the
similarity between this line pattern and that of a current loop.
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