Download Physics 272: Electricity and Magnetism

Physics 272: Electricity and
Magnetism
Mark Palenik
Tuesday, July 3rd
Again. . . midterm
• Remember, today is the last lecture (with me,
or for the week, with anyone)
• Wednesday, no class
• Midterm on Thursday
Topics for today
• Force on a moving charge by the magnetic
field
• Force of a current carrying wire by the
magnetic field
• Atoms as magnetic dipoles
• Interaction of dipoles with the magnetic field
• Hall effect
First. . .
Magnetic field of a Wire Loop
< −𝑅 cos 𝜃 ,−𝑅 sin 𝜃 ,𝑧>
• ∆𝑙 × 𝑟 =< −𝑅 sin 𝜃 , 𝑅 cos 𝜃 , 0 > ∆𝜃 ×
𝑟
2
2
• =< 𝑅 cos 𝜃 𝑧, 𝑅 sin 𝜃 𝑧, 𝑅 >/𝑟 → 𝑅
𝜇0
1
𝜇0 ∆𝑙
𝜇0
∆𝑙
• ∆𝐵 = ∆𝑙 × 𝑟 2 =
=
4𝜋
𝑟
4𝜋 𝑟 3
4𝜋 (𝑧 2 +𝑅 2 )3/2
• 𝑑𝑙 × 𝑟 = 𝑅2 𝑑𝜃
• 𝐵=
2𝜋 𝜇0 𝑅 2 𝑑𝜃𝑧
0 4𝜋 (𝑧 2 +𝑅 2 )3/2
m0 2p R 2 I
Bz =
4p ( R 2 + z 2 )3/2
Magnetic force (Lorentz force)
• Magnetic fields act on moving charges
• Lorentz force on an object is 𝐹𝐿 = 𝑞𝑣 × 𝐵
• In the presence of an electric and magnetic field,
𝐹 = 𝑞(𝐸 + 𝑣 × 𝐵)
iClicker question: Magnetic force
• The fact that 𝑣 × 𝐵 appears in the equation
for force means that
a)
b)
c)
d)
e)
F is parallel to v
F is parallel to B
F can have a component parallel to v or B
F is perpendicular to v and B
F is only perpendicular to v
iClicker question: Work done by
magnetism
𝑏
𝐹
𝑎
• Recall the definition of work (𝑊 =
• Does the magnetic field do work?
∙ 𝑑𝑥 )
a) Yes, because the direction of F is not important
b) No, because F is perpendicular to B
𝑏
𝐹
𝑎
c) Yes, because even if 𝐹 ∙ 𝑑𝑥 = 0,
not be zero
d) No, because F is perpendicular to v
∙ 𝑑𝑥 may
Magnetic fields do no work (usually)
• Since the force due to the magnetic field is always
perpendicular to velocity, work=0
• Acceleration is always perpendicular to v. What shape
does that sound like?
Charged particles in a B field often
move in a circle (they will if the field
is 1) uniform and 2) perpendicular to
v)– Acceleration and velocity are
perpendicular
Also, magnitude of acceleration is wv
(constant * v)
iClicker: Which field is bigger
• The same charge is place in two different uniform
magnetic fields that point into the page.
• The charge is initially moving to the right with a speed
v in both situations.
• Which circle is the path of the charge in the STRONGER
field?
a)
b)
Magnetic force on a wire
• Remember, “I” is a sort of moving charge density.
𝑞𝑣
essentially ,
𝐿
• It is
the amount of moving charge per
unit length.
• We can find force per unit length on a wire
• 𝐼×𝐵 =
• 𝐹=
𝑞𝑣×𝐵
𝐿
=
𝐹
𝐿
B
𝐼 × 𝐵𝑑𝑙
• B is the external field
– NOT the field produced by the wire
F on this small
segment of wire is
IB into the page
iClicker: Magnetic force between 2
wires
• Two wires run parallel and have currents that
run in the +y direction. Do they:
y
z
I
a) Attract
b) Repel
I
x
iClicker: Magnetic force between 2
wires
• Two long wires with length L carry a current in the +y
direction
• Keep in mind the magnitude of the magnetic field
𝜇 2𝐼
produced by a long wire is B= 0
4𝜋 𝑟
• What is the magnitude of the force between the wires?
a)
b)
c)
d)
𝜇0 2𝐼 2
4𝜋 𝑟
𝜇0 2𝐼 2 𝐿
4𝜋 𝑟
𝜇0 2𝐼𝐿
4𝜋 𝑟
𝜇0 2𝐼 2 𝐿2
4𝜋 𝑟
I
I
When magnetic fields CAN do work
• Take a dipole, like an electron, and place it in a
magnetic field
• The energy of the dipole is −𝜇 ∙ 𝐵
• When 𝜇 is the same direction as 𝐵, energy is minimized
• When 𝜇 is the opposite direction of 𝐵, energy is
maximized
• At any other point there is a torque on the dipole—the
dipole wants to go to the lowest energy orientation.
How will a dipole behave in a uniform
field?
• A magnetic dipole points at a 45 degree angle in the xy plane
• It is placed in a uniform magnetic field that points in the +y
direction
• What happens to the dipole?
y
z
a)
b)
c)
d)
e)
x
There is a torque that rotates it clockwise
There is a torque that rotates it counter clockwise
Nothing
A force pushes it right
A force pushed it left
How will a dipole behave in a nonuniform field?
• Remember, the energy of a dipole is −𝜇 ∙ 𝐵
• A magnetic dipole pointing in the +y direction is placed in a
non-uniform magnetic field that points in the –y direction
decreases along the x axis.
y
• Which direction is the force on the dipole?
z
a)
b)
c)
d)
e)
+x
B m
-x
+y
-y
There is no force, because it is a magnetic field
x
Atoms as magnetic dipoles
• Electron spin and orbital angular momentum give them
dipole moments.
• A single atom produces a weak field
• Several atoms can be aligned by a strong external field
• Electric/fermi interactions can hold them in this
alignment
Bar Magnet
• A bar magnet is composed of a bunch of spin aligned
atoms. It is a collection of dipoles
• The field is not exactly the same as a dipole (when very
close), but looks a lot like a dipole from far away
• Field lines leave the north end and enter the south
Other dipoles, earth, compass needle
Same as a compass needle
N
S
The earth is backwards!
Hall effect
• Let’s think about two problems relating to the
picture below that will give insight into the
hall effect
+V
-V
• Since the positive voltage is on the left,
conventional current runs to the right.
Electron current runs to the left.
iClicker: Hall effect
• Assume we add a magnetic field that points out of the
page. Which way would the magnetic field push the
electrons as they move through the metal block?
Metal block
a)
b)
c)
d)
Up
Down
Left
Right
+V
-V
iClicker: Hall effect
• The magnetic field still points out of the page. If the idea
that positive charges moved were correct (so positive
charges move in the direction of conventional current),
which way would the magnetic field push the positive
charges as they move through the metal block?
Metal block
a)
b)
c)
d)
Up
Down
Left
Right
+V
-V
Hall voltage
• If electrons are pushed down, negative charges accumulate
at the bottom. Electric field points down in the block, the
bottom is at a lower voltage
Metal block
+V
E
----------------
-V
• If positive charges are pushed down, the voltage would be
higher at the bottom
Metal block
+V
E
+++++++++
-V
Note: conductor is
not at equilibrium,
which is why we
can have an E field
inside.