Lecture 11.1 :
Ampere’s Law and Magnetic Force
Lecture Outline:
Ampere’s Law
Solenoids
Magnetic Force on Moving Charge
Textbook Reading:
Ch. 32.6 - 32.7
March 26, 2013
1
Announcements
•Homework #9 due on Monday, April 1 in Mastering Physics.
•Exam #3 is next week, April 4.
Covers Ch. 30-32.
•Review session for Exam #3 will be offered on Tuesday, April 2,
from 7-9pm in Stolkin.
‣NOTE:
If you are in the Tue. lab section (from 6:30-8:30pm), you can
go to another lab section next week, or you can join the review session
late. Apologies for this overlap (again).
•Advertisement: Wali Lecture tonight at 5pm in Gifford Auditorium:
Conversing with the Dalai Lama: 25 years of dialogue with science - Prof. Arthur Zajonc
2
Last Lecture...
Notice that a current loop has a directionality. Magnetic field
flows in one side, and out through the other, just like a magnet.
The current loop is a magnet!
3
Last Lecture...
Recall the electric dipole...
p� = qs
Dipole moment, points
from negative to positive
charge.
p
� dipole
E
1 p�
≈−
4π�0 r3
Electric field along dipole
axis, far from dipole.
4
Last Lecture...
We can also define a magnetic dipole moment
Bloop
IR2
µ0
=
2 (z 2 + R2 )3/2
z >> R
Bloop
µ0 IR2
µ0 2(πR2 )I
µ0 2AI
≈
=
=
3
3
2 z
4π
z
4π z 3
µ
� ≡ AI
Magnetic dipole moment points
from south pole to north pole.
Bdipole
5
µ0 2�
µ
=
4π z 3
Ampere’s Law
We can use the Biot-Savart law to calculate magnetic field of any
distribution of moving charges. In practice, this is difficult (like
using Coulomb’s Law to calculate E-field of arbitrary charge
distributions).
Sum/Integrate B along the line
�
Line passing through a
magnetic field.
k
6
� k · ∆�sk →
B
�
f
i
� · d�s
B
Ampere’s Law
Calculate the same integral around the circumference of a circle
with radius d, surrounding a wire carrying current I
�
�
B is constant/parallel to
ds everywhere on circle!
� · d�s = B(2πd)
B
�
7
� · d�s
B
µ0 I
�
Bwire =
2π d
� · d�s = µ0 I
B
Ampere’s Law
Result from previous slide turns out to be general:
•Independent of the shape of the curve around the current.
•Independent of where the current passes through curve.
•Depends only on the total current passing through the area
enclosed by the integration path.
Ampere’s Law
Note that the integral here is around a closed loop, and not over a closed surface like in Gauss’s Law.
8
Ampere’s Law
Must be careful to determine which currents passing through the
integration path are positive and which are negative.
Use right-hand rule to
determine sign of the
current(s).
What is Ithrough for this loop?
9
Clicker Question #1
For the path shown,
A.
B.
C.
D.
0.
µ0(I1 − I2).
µ0(I2 − I1).
µ0(I1 + I2).
10
Ampere’s Law
Example: What is the magnetic field inside and
outside of a current-carrying wire of radius R ?
11
Solenoids
A solenoid is a helical coil of wire with current I
passing through each loop in the coil.
12
Solenoids
What is the magnetic field inside the solenoid?
13
Clicker Question #2
Solenoid 2 has twice the diameter, twice the length, and twice as
many turns as solenoid 1. How does the field B2 at the center of
solenoid 2 compare to B1 at the center of solenoid 1?
A.
B.
C.
D.
E.
B2 = B1/4.
B2 = B1/2.
B2 = B1.
B2 = 2B1.
B2 = 4B1.
Same number of
turns/length!
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Solenoids
Real Solenoids are Very Useful
Magnetic Resonance Imaging
3 Tesla fields are common.
15
Magnetic Force on Moving Charge
Magnetic field exerts a force on a current!
Turning off the current in either wire will make the
attractive/repulsive force vanish.
16
Magnetic Force on Moving Charge
Magnetic Force also depends on how moving charge’s
velocity is oriented relative to a magnetic field.
17
Clicker Question #3
The direction of the magnetic force on the proton is
A.
B.
C.
D.
E.
To the right.
To the left.
Into the screen.
Out of the screen.
The magnetic
force is zero.
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Magnetic Force on Moving Charge
Example: What’s the magnetic force on the electron if it’s 1.0cm
above a 10A current, and it’s traveling at 1.0x107 m/s to the right?
Follow-up: What happens to the electron as it travels? Does it
speed up or slow down?
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Magnetic Force on Moving Charge
Cyclotron Motion is result of a particle moving
perpendicular to a uniform magnetic field.
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Magnetic Force on Moving Charge
184-inch cyclotron
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Magnetic Force on a Moving Charge
•Modern cyclotrons can produce radioactive tracers used in medical
applications, and can also be used directly to treat tumors.
•Technetium-99 is the most common isotope used in nuclear medicine.
•Technetium produced by radioactive decay of a rare isotope of
Molybdenum (Mo-99...which comes from fission of U-235) or ~20 MeV
proton bombardment of a more common isotope (Mo-100).
Technetium is radioactive and decays, producing
easily detectable photons. Allows function of kidneys
and other organs to be studied.
Protons accelerated in cyclotrons can be targeted at
tumors within the body (right), causing less collateral
damage than X-ray methods (left).
22
Reminders
•Homework #9 is due Monday (April 1).
•Exam #3 next week (April 4), and a review session
on April 2 from 7-9 pm in Stolkin.
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