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MTE2 Wed 26, at 5:30-7:00 pm
Ch2103 and SH 180
Contents of MTE2
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Work of the electric force and potential energy
Electric Potential and Field
Capacitors and capacitance
Current and resistance, Ohm’s law
DC Circuits and Kirchoff’s laws
RC circuits
Lorentz force and motion of charge in a magnetic field
Biot and Savart
No Ampere’s law, No magnetic properties of matter
Alternate Exams will be:
Wed 26
2:30-4:00pm Lab room (Ch 3254)
6:00-7:30pm Lab room (Ch 3254 )
Study chapters 27-32 (no 32.6, 32.10, no
32.8 forces between wires)
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This week Honor Lecture
Magnetic Fields
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Susan Nossal: Climate change
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A vector quantity (B)
compass needle traces B field lines and points towards N
B-field lines start in N and go to S
they do not start or stop (no magnetic monopoles
Electric dipole
Iron filings show pattern of B-field lines
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Earth’s A Magnet!
Electric vs Magnetic Field Lines
N geographic pole almost at
magnetic S pole
!S geographic pole almost at
magnetic N pole
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A the N pole of a
compass needle is
attracted toward the
geographic N pole (S
pole of the B-field)
Similarities
! Density gives strength
! Arrow gives direction
! Leave +, North
! Enter
-, South
Differences
! Start/Stop on electric charge
! No Magnetic Charge !
lines are continuous!
+
Charged particle in a B-field:
Lorentz force
The magnetic force
r
r r
F = qv " B vector ‘cross product’
acts only a moving
charged particle
Units: N = C x m/s x Tesla
Effect of uniform magnetic field
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Effect of uniform B-field on charged particle
! If charged particle is not moving - no effect
! If particle is moving: force perpendicular
to both field and velocity
! the charge sign must be accounted for
FB = q vB sin "
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F is zero when v and B are parallel
or antiparallel
! F is a maximum when v and B are
! perpendicular
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The magnetic force NEVER does work on the particle.
Magnetic Force and Work
Quick Quiz
Magnetic force perpendicular to the particle displacement
=> the magnetic force does not do work!
The three charges below have equal charge and speed,
but are traveling in different directions in a uniform
magnetic field.
B field can alter direction of velocity,
but not the speed or the kinetic energy
Which particle experiences the greatest magnetic force?
B
x x x x x x
B
x x x x x x
v
x x x x x x
B
########
v
# # # # #v# # #
F
q
q
F
q
F=0
A) B) C)
1 2 3
D)
Same
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B
F = q v B sin($)
2
1
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Mass spectrometer and v selector
Cyclotron Motion
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F " to v and B
x x x x x xx x x x x x
x x x x x xx x x x xv xB
v
F
F
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particle path is a circle of
radius R.
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a = v2/R is the centripetal
acceleration
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R depends on m, q, v, B
q
R
In how much time does the particle complete an orbit?
F = ma
v qB
=
R m
2#
2#m
"=
, T = period =
T
qB
" = angular velocity =
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Experiment in Lab: motion of
charges in B-field
Magnetic Force on a Current
e-
1) Turn on electron ‘gun’
Force on each charge
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Force on length ds of wire
B
R
1 2
mv = q"V
2
2) Turn on magnetic field B (Helmotz coils)
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%V
qv " B
qv " Ids
dF = Ids " B
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‘gun’
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3) Measure electron charge/mass as a function
of !V, R and B
m 2v 2
q
m2
q
R 2 = 2 2 = 2 "V 2 2
v 2 = 2 "V
q
B
m
q
B
m
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FB = I L " B
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q 2"V
=
m R 2B 2
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Torque on a current loop
‘Homopolar motor’
A homopolar motor has a B-field along
the axis of rotation and an electric current
produced by the battery that moves
radially through the magnet.
B
Ids
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The resulting Lorentz force in the
tangential direction produces a torque in
the magnet, which is free to rotate with
the attached screw. The name
homopolar refers to the absence of
polarity change.
Consider current loop in magnetic
field perpendicular to the plane of
the loop => Zero net force
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Force on top path cancels
force on bottom path (F = IBL)
x
x
x
Fx
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
F
x
x
x
x
x
i
x
x
x
x
x
B
x
x
x
x F
x
F
Force on right path cancels force on left path. (F = IBL)
B
F
If plane of loop is not
to B-field, there
x
is a torque on the loop (force couple
as for electric dipole in E-field)
FB = I L " B
F
F
.
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Calculation of Torque
• Square loop has width W and length L (into
the screen). The torque is given by:
r r r r r
" = r1 # F + r2 # F
$w
'
" = 2& F sin # )
%2
(
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rxF
F = IBL " # = IBLw sin $ = IBAsin $
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A = wL = area of loop
Potential Energy of Magnetic Dipole
B
I
r
$ x 1
w
F
F
r2 .
B
$ x
magnetic dipole moment
direction: " to plane of the loop
µ=AI
according to RH rule (hands curls on
current)
• Note: if loop consists of N turns, µ = N A I
w
F
$
µ
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Work must be done to change the orientation of a dipole
(current loop) in the presence of a magnetic field.
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Define a potential energy U corresponding to this work
(U=0 when magnetic dipole moment orthogonal to B-field
= equilibrium position corresponding to max torque)
B
F
.
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x
w
F
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µ
$
F
.
Quick Quiz
Surprise!
Which configuration has the greatest potential energy?
µ
B
µ
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x
B
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B
!
µ
x
Electric current
produces magnetic field
Current (flow of electric charges ) in wire produces B-field.
The magnetic field aligns compass needle
2 kind of magnetism due to permanent magnets and currents or
the same?
x
Current
&=0
& = µB
&=0
U = -µB
U=0
U = µB
A)
B)
Magnetic
field
C)
1819 Hans Christian Oersted
Magnetic field from a current
Iron filings align with
magnetic field lines
Current loop as magnetic dipole
A current loop is an electromagnet: it behaves like a permanent
magnet. Two current loops with current in the same direction
attract like 2 magnets (since the opposite poles are closer)
Magnetic field loops around the
current. (Right hand rule)
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Field from a circular loop
Biot-Sarvart Law
For a single charge in motion
ur
B=
For a wire with current
µ0 qv # ur
4" r 2
At the center of the loop:
dB =
B out of page
I
ds
!
µ ds # ur
dB = 0 I
4"
r2
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! Each element of current
produces a contribution to
the magnetic field.
permeability of free
space
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B=
!
B=
µ0 = 4 " #10$7 Tm / A = 1.257 #10$6 Tm / A
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µ0 Idlsin # # = 90 o
µ Idl
$ $$% dB = 0 2
4" R 2
4" R
qv " Ids
r $
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µ0 I
4" R 2
# dl
µ0 I
µI
2"R = 0
2
4" R
2R
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Happy spring break to all!
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