Thursday, 28 July 2015
Announcements
• All grades so far are posted – do check!
• Midterm results will not be available before Tuesday.
• Homework
– Due date
– Comments – see website
The magnetic field at the position P points…
A. Into the page.
B. Out of the page.
C. Down.
D. Up.
The outline
Past:
• Electric fields
• sources: charge => Coulomb’s law, Gauss
• Effect: F=qE – force on charge
• Electric potential
• point charge, E-field <-> potential
• Effect: U=qV – energy of charge; Kirchhoff’s
loop law!
• Potential in circuits+ circuit stuff…
Today and Monday:
• Magnetic Fields
• Sources: magnetic dipole, moving charge,
current => Bio-Savart law; Ampere’s law
• Effect: F=qvxB
• Math: cross product and RHR
Future:
• Changing magnetic field
• Electromagnetic induction
• Electromagnetic waves
A charged rod is held next to a
compass. Does the compass
needle…
A. Rotate Clockwise
B. Rotate Counterclockwise
C. Not react
Electric and magnetic fields/forces
Electric Field:
• Long-range force
• Positive and negative
charge
• Unlike charges attract; like
charges repel
• Charged object can induce
and electric dipole
Long-range force – but not affecting
everything…
Electric and magnetic fields/forces
Electric field
• Long-range force
Magnetic field
• Long-range force
If a bar magnet is cut in half, you
end up with
Electric and magnetic fields/forces
Electric field
• Long-range force
• Positive and negative
charge
Magnetic field
• Long-range force
• North and south pole
Electric and magnetic fields/forces:
Similarities
Electric and magnetic fields/forces
Electric field
• Long-range force
• Positive and negative
charge
• Unlike charges attract;
like charges repel
Magnetic field
• Long-range force
• North and south pole
• N and S attract; like
poles repel
If the bar magnet is flipped over and the south pole is
brought near the hanging ball, the ball will be
A. Attracted to the magnet.
B. Repelled by the magnet.
C. Unaffected by the
magnet.
D. I’m not sure.
Electric and magnetic fields/forces
Electric field
Magnetic field
• Long-range force
• Long-range force
• Positive and negative
• North and south pole
charge
• Unlike charges attract; like • N and S attract; like poles
repel
charges repel
• Magnet can induce a
• Charged object can
induce and electric
magnetic dipole
dipole
Magnetic Force on a Compass
 The figure shows a compass
needle in a magnetic field.
 A magnetic force is exerted
on each of the two poles of
the compass, parallel to for
the north pole and opposite
for the south pole.
 This pair of opposite forces
exerts a torque on the
needle, rotating the needle
until it is parallel to the
magnetic field at that point.
Electric and magnetic fields/forces
Electric field
• Long-range force
• Positive and negative
charge
• Unlike charges attract; like
charges repel
• Charged object can induce
and electric dipole
• Electric dipole points in
the direction of the
electric field
Magnetic field
• Long-range force
• North and south pole
• N and S attract; like poles
repel
• Magnet can induce a
magnetic dipole
• Magnetic dipole points
in the direction of a
magnetic field
How did we know if there is a field?
• Electric field?
• Put a positive point charge
into the region.
•
Magnetic field?
• Place a magnetic dipole
into the region.
What causes magnetic field
• Magnetic dipole, and ….
• Natural - lodestone
What causes magnetic field?
• Magnetic dipole
• Naturally magnetized – lodestone, the “leading
stone”(magnetite)
–
Also pyrrhotite – weakly magnetic
The SI unit of magnetic
field strength is the
tesla, abbreviated as T:
1 tesla = 1 T = 1 N/A m
What causes magnetic field?
• Magnetic dipole
• And...
Moving charges cause magnetic fields

  0 qv  rˆ
B
2
4 r
0 = 4 × 10 T m/A
-7
Biot-Savart Law
Units of magnetic field: 1T (tesla)
Vector cross product
  
C  A B
A) I am comfortable working with this
operation
B) I remember this, but need practice
C) Perhaps I have seen it, but I will need
help understanding
D) I do not remember seeing this, and it
does not make sense.
Important ones
xˆ  yˆ  ??
A) xˆ
B )  xˆ
C ) yˆ
D)  yˆ
E ) zˆ
F )  zˆ
Important ones
xˆ  yˆ  ??
A) xˆ
B )  xˆ
C ) yˆ
D)  yˆ
E ) zˆ
F )  zˆ
Vector cross product: Math Review

C=
A×
B
C x  Ay Bz  Az B y

C
C=ABsin(α),
RHR for direction

B
α

A
Right-hand Rule: Review

C=
A×
B

C

B
α

A
Right-hand Rule: Review

C=
A×
B

C

B
α

A
Right-hand Rule: Review

C=
A×
B

C

B
α

A
The Source of the Magnetic Field: Moving
Charges
The magnetic field of a
charged particle q moving
with velocity v is given by
the Biot-Savart law:
The constant 0 in the Biot-Savart law is called the
permeability constant:
0 = 4 × 10-7 T m/A = 1.257 × 10-6 T m/A
Moving charges cause magnetic fields


 0 0 v  rˆ
1
B
q
2
4 0
r

E
1
rˆ
q 2
4 0 r
QuickCheck 32.5
What is the direction of the magnetic field at the
position of the dot?
A.
B.
C.
D.
E.
Into the screen.
Out of the screen.
Up.
Down.
Left.
Slide 32-54
Example
• Proton moving along the
+z-axis with speed
100m/s. What is the
magnetic field due to the
proton at (2,1,0)m, at the
time when the proton
goes through the origin?
Magnetic Field of a Moving Positive Charge
 The right-hand rule for finding
the direction of due to a
moving positive charge is
similar to the rule used for a
current carrying wire.
 Note that the component of
parallel to the line of motion is
zero.
Slide 32-46
What causes magnetic field?
• Magnetic dipole
• Moving Charges
• And...
Current is moving charges…
• Hans Christian Oersted – discovered magnetic fields due
to a wire in a lecture demo, 1819
Electric currents cause magnetic fields.
Use right hand rule:
I
B
Tactics: Right-Hand Rule for Fields
Different RHR!
Slide 32-43
Moving charges cause magnetic fields

  0 qv  rˆ
B
2
4 r
Δs
A
I
The Magnetic Field of a Current
Slide 32-58
Superposition of Magnetic Fields
 Magnetic fields, like electric fields, have been found
experimentally to obey the principle of superposition.
 If there are n moving point charges, the net magnetic
field is given by the vector sum:
Moving charges cause magnetic fields

  0 qv  rˆ
B
2
4 r
Δs
A
I
Problem-Solving Strategy: The Magnetic Field of
a Current
Problem-Solving Strategy: The Magnetic Field of
a Current
Slide 32-63
A wire of length L...
ds
I
A
θ
d


 0 ds  rˆ
B
I 2
4
r
Direction B
away from us
 0 sin( )ds
B
I
2
4
r
A wire of length L...
ds
I
A
θ
yP
Direction B
away from us
μ 0 Iy P
B=
4π
−L/ 2
∫
L/2
dx
3
r
r =√ x + y
2
2
p
See Appendix A page 3
xdx
x
∫ (x2 +a2 )3/2 = 2 2 2
a √ x +a
The magnetic field due to an infinitely long wire...
The magnetic field of a long, straight wire carrying current I at a distance d
from the wire is:
Slide 32-59
QuickCheck 32.6
Compared to the magnetic field at point A, the magnetic
field at point B is
A. Half as strong, same direction.
B. Half as strong, opposite direction.
C. One-quarter as strong, same
direction.
D. One-quarter as strong, opposite
direction.
E. Can’t compare without knowing I.
Slide 32-60
A long straight wire passes
through a uniform magnetic
field, B. The net magnetic
field at point 3 is zero.
a) Show the direction of
current in the wire.
b) Draw vector diagrams at
points 1, 2 and 4 to
determine the net magnetic
field at these points in terms
of B.
Biot-Savart
Field due to short segment of wire:

dl


r

  0 I dl  rˆ
dB 
2
4 r
Application Biot-Savart
Field at center of loop, radius R.
0 I dl
dB 
2
4 R
I
dl
B
 0 I 2R  0 I
B

2
4 R
2R
The Magnetic Field of a Current
The magnetic field of a long, straight wire carrying
current I at a distance d from the wire is:
The magnetic field at the center of a coil of N turns
and radius R, carrying a current I is:
Slide 32-59
QuickCheck 32.7
The magnet field at point P is
A. Into the screen.
B. Out of the screen.
C. Zero.
Slide 32-72