Download Physics 272: Electricity and Magnetism

Physics 272: Electricity and
Magnetism
Monday July 2nd
Mark Palenik
Midterm
• Reminder: Thursday at 9:50, 2 hours
– We will go over answers afterward
• Tomorrow is the last lecture (Wednesday July
4th no class)
• No recitation Thursday
Midterm topics
•
•
•
•
Electric field of a point charge
Electric field of dipoles (on axis and perpendicular axis)
Polarization of materials
Electric field and conductors
– Field inside a conductor
– Field produced by a conductor
•
•
•
•
Electric field of a uniformly charged sphere (inside and outside)
Electric field of a rod (short, infinite)
Electric field of a capacitor
Electric potential and potential difference
– Of a generic electric field
– Of a point charge
– Go back and forth between E and V
•
•
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Electrostatic potential energy
Energy stored in the electric field
Conventional current and electron current
Biot-Savart law
– Moving charge
– Currrent element
•
Cross products
Topics for today
• Magnetic field of a current loop
• Field of magnetic dipoles
– Electric vs. magnetic dipoles
– Current loops and dipoles
• Atoms as dipoles
• Also: A bar magnet is a magnetic dipole, but
we’ll talk about that more tomorrow
Magnetic field of current distribution
• Remember Biot-Savart law:
μ0
4𝜋
𝐼×𝑟
𝑑𝑙
2
𝑟
1. Cut up the current distribution into pieces
and draw B
2.Write an expression for B due to one piece
3.Add up the contributions of all the pieces
4.Check the result
We will also be exploiting some symmetries of
the object, so lets think about those
Recall again. . . Right hand rule
• Thumb in direction of conventional current,
fingers wrap in direction of the wire
• Electron current and conventional current
point in opposite directions.
iClicker: electron current
• Which direction does the electron current in
the wire point?
a)
b)
c)
d)
e)
North
South
East
West
The current is zero
iClicker
• What is the direction of the magnetic field at
the x?
y
a)
b)
c)
d)
e)
+y
-y
+z
-z
0
z
x
iClicker: Ring
• Remember what we said on the previous slide. If
the conventional current is running clockwise in
the loop, which way does the magnetic field point
at the center?
a)
b)
c)
d)
e)
It is zero
+y
-y
+z
-z
y
I
z
I
I
x
I
x
A third right hand rule
• Wrap fingers in the direction of the current
loop. Thumb points in the direction of B.
I
I
I
x
I
Magnetic Field of a Wire Loop
Step 1:
Cut up the distribution into pieces
Dl = R cos (q + dq ) , Rsin (q + dq ) ,0 - R cosq , Rsin q , 0 → −𝑅 sin 𝜃 , 𝑅 cos(𝜃) 𝑑𝜃
r = 0,0, z - R cosq , R sin q ,0
Make use of symmetry!
Need to consider only Bz due
to one dl
Magnetic field of a Wire Loop
• ∆𝑙 × 𝑟 =< −𝑅 sin 𝜃 , 𝑅 cos 𝜃 , 0 > ∆𝜃 ×< 𝑅 cos 𝜃 , 𝑅 sin 𝜃 , 𝑧 >
• =< 𝑅 cos 𝜃 𝑧, 𝑅 sin 𝜃 𝑧, −𝑅2 >→ −𝑅2
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 Field of a Wire Loop
m0 2p R 2 I
Bz =
4p ( R 2 + z 2 )3/2
Step 4:
Check the results
æ T × m ö ( m × A)
=T
çè
÷ø
A (m )
2
 units:
2 3/2
 direction:
Check several pieces with
the right hand rule
Note: We’ve not calculated or shown the “rest” of the magnetic
field
Current carrying ring vs. charged ring
• Compare the electric field at the center of a
uniformly charged ring to the magnetic field in
a current-carrying ring. What do you notice
about the field strength at the center of the
ring?
• What happens as you pass through the center
of a current carrying ring vs. charged ring?
Magnetic dipole
• 𝐵𝑧 =
𝜇0
2𝜋𝑅 2 𝐼
4𝜋 𝑅 2 +𝑧 2 3/2
• z>>R, so
𝑅2
+
take the limit as we get very far away
𝑧 2 3/2
→
𝑧3
and
𝜇0
2𝜋𝑅 2 𝐼
4𝜋 𝑅 2 +𝑧 2 3/2
• What does this look like?
Like an electric dipole!
1 2p
Ez 
4 0 z 3
p  sq
0 2R 2 I
Bz 
4 z 3
→
𝜇0 2𝜋𝑅 2 𝐼
4𝜋 𝑍 3
 of a loop
• From far away, Bz =
𝜇0 2𝜋𝑅 2 𝐼
4𝜋 𝑍 3
• And from an electric dipole, Ez =
1 2𝑃
4𝜋∈0 𝑍 3
• For electric dipoles, we called the dipole moment P.
• For magnetic dipoles, the dipole moment is called 
• What is  for a loop?
𝜇0 2𝜋𝑅 2 𝐼
4𝜋 𝑍 3
2
• Bz =
• 𝜇 = 𝜋𝑅 𝐼
=
𝜇0 2𝜇
4𝜋 𝑍 3
iClicker question
• Which axis do we have to flip on this current
ring to reverse the direction of the magnetic
field everywhere?
y
a) X
b) Y
c) Z
z
x
Reflection of Current Ring
y
y
x
x
z
z
B µ v ´ r̂
B ® -B
Magnetic vs. Electric dipoles
• How do the direction of the electric and magnetic field
lines compare in the two dipoles below?
• Which way do we have to flip a magnetic dipole?
• Which way do we have to flip an electric dipole to reverse
the direction of its field at every point?
-q
+q
N Current loops
• If we have a bunch of loops sitting on top of
each other, we can usually pretend they’re all
in exactly the same place.
• Field from N loops = N*Field from one loop
• Bz =
𝜇0 2𝜋𝑁𝑅 2 𝐼
4𝜋 𝑍 3
I
B
iClicker  of N loops
• What is the magnetic moment, , of a current
carrying ring with N loops, radius R, and
current I?
a)
b)
c)
d)
2NR2I
NR2I
0NR2I
20NR2I
Atoms as dipoles
• Remember, current loops are dipoles
• Shrink the loop down and you get a point dipole
• A point dipole doesn’t exactly have current, since there is
no motion (single point)
• Electrons can still have angular momentum.
• Electrons have “spin” angular momentum and behave as
point dipoles.
• Can only know one component which can take on 2 values
Atoms as dipoles
• Electrons can also have orbital angular
momentum around the nucleus
• You can think of it like a classical orbit
(current)
• This also produces a magnetic field, like a
regular current loop.