Download Physics 227: Lecture 15 Magnetic Fields from wires

Physics 227: Lecture 15
Magnetic Fields from wires
•
Lecture 14 review:
•
•
•
•
•
•
Force on current/wire loops = 0, in
uniform field
Current loop magnetic moment μ = I A.
Torque on current loops: τ = μxB.
Application: DC motor.
Induced magnetic moments pull
ferromagnetic materials into high field
regions.
Hall effect: current carriers pushed to
side of wire.
Forces between parallel wires
Thursday, October 27, 2011
Field from a Moving Charge
We have already discussed, and seen in
demos, how the magnetic field from a wire
carrying a current is circular. A current is a
set of charges, so it is not surprising that
the field of a moving charge and a current
are the same shape.
Use the RH hand to determine the direction.
Thursday, October 27, 2011
iClicker: What is the Magnetic Field
Direction?
P
+Q
v
When the particle is in the position shown to the left, moving to
the right, what is the direction of the magnetic field at point P?
A. Down to the right.
B. Up to the right.
C. Out of the board.
D. In to the board.
E. Vertical, up or down.
Thursday, October 27, 2011
See previous page, or use RH rule.
What is the Magnetic Field Magnitude?
µ0 q�v × r̂
�
B(�r) =
4π r2
Using cylindrical coordinates,
you can see the formula and
picture agree: the velocity is in
the z direction, the radial
coordinate is radial, so the Bfield is in the azimuthal (θ)
direction.
μ0 = 4πx10-7 Tm/A or Ns2/C2
Thursday, October 27, 2011
Note that for a single moving
particle, the field is most
intense near it, and decreases
for the same distance from the
wire when further from the
particle, as 1/r2 is smaller.
What is the Force Between Moving Charges?
The lower + charge, moving in the
+x direction, generates a magnetic
field in the +z direction at the
position of the upper + charge. The
magnetic force on the upper charge
is FM = qvB in the +y direction.
There is also an electric field from
the lower charge at the position of
the upper charge, leading to an
electric force FE = qE in the +y
direction.
What is the relative magnetic of
the electric and magnetic forces?
Thursday, October 27, 2011
iClicker: What is the Relative Magnitude
of the Electric and Magnetic Forces
between two Equal Charges?
A. Electric force is always greater.
B. Magnetic force is always greater.
C. The two forces are always equal.
D. The electric force is never less.
E. The magnetic force is never less.
Answer follows in two slides.
Thursday, October 27, 2011
iClicker: What is the Relative Magnitude
of the Electric and Magnetic Forces
between two Equal Charges?
A. Electric force is always greater.
B. Magnetic force is always greater.
C. The two forces are always equal.
D. The electric force is never less.
E. The magnetic force is never less.
A note on ``relativity’’: What happens if you are
moving along to +x with the lower particle?
Thursday, October 27, 2011
What is the ratio of Electric to Magnetic Force?
µ
q�
v
×
r̂
0
� r) =
B(�
4π r2
2 2
µ
qv
µ
q
v
0
0
� = qv
|F�M | = |q�v × B|
=
4π r2
4π r2
q2
FE =
4π�0 r2
FM
= µ0 �0 v 2
FE
What is μ0ε0v2? From their numerical
values, you can see that μ0ε0 = 1/(3x108)2.
Later on we will learn that the speed of
light is c2 = 1/μ0ε0, so μ0ε0 = v2/c2 < 1
always, and FM < FE for v < c.
Thursday, October 27, 2011
What if the Velocities are Parallel, Rather than
Antiparallel?
All the equations are
the same, but some
directions reverse.
Thursday, October 27, 2011
Change of Pace - Paramagnetism Demo
Ferromagnets - large magnetization aligned to applied
field.
Paramagetics - small magnetization aligned to applied
field, attracted into strong field region.
Diamagnetics - small magnetization anti-aligned to
applied field, repelled out of strong field region.
Thursday, October 27, 2011
What is the Magnetic Field of a Current?
µ0 q�v × r̂
�
B(�r) =
4π r2
We need to go from a single charge to integrating over many
charges, for now all the same magnitude and moving along the
same straight section of wire.
�
q�
v
×
r̂
µ
Id
l
×
r̂
µ
0
0
� r) = nAdl
=
dB(�
4π r2
4π r2
``Biot-Savart’’ Law.
To calculate the field from an infinitely long wire, we need to do an
integral ...
Thursday, October 27, 2011
The Magnetic Field of an Infinitely long
Wire / Current?
�
q�
v
×
r̂
µ
Id
l
×
r̂
µ
0
0
� r) = nAdl
=
dB(�
4π r2
4π r2
distance from wire = y
θ
x=-a
B=
�
x=0
a
−a
µ0 Idx
4π x2 + y 2
�
x=a
�
y
x2 + y 2
Do not need to worry about components
of B, B is always out of page for all x
Thursday, October 27, 2011
�
→a→∞
µ0 I
2πy
Usually written
B = μ0I/2πr
sinθ of cross product
The Magnetic Field of Two Infinitely long
Wires / Currents?
Magnetic fields add, just as electric fields add, so BTotal = B1 + B2.
Usually written B = μ0I/2πr
Thursday, October 27, 2011
Magnitude of Force Between
Parallel Conductors
Consider two wires carrying current
in the same direction. We have
already seen the force is attractive.
The magnetic field at one wire from
the other is B = μ0I/2πr.
The force on a length L of the 2nd
wire is F = ILxB = μ0I2L/2πr, or the
force per unit length is
F/L = μ0I2/2πr.
Thursday, October 27, 2011
Magnetic Field of a Wire Loop - on Axis
As we go around the loop,
the y components of the
field rotate and cancel. Only
the x component survives.
Note that nothing in the
integral depends on θ - it is
an easy integral to do!
If you have a coil with N
loops, multiply by N.
Bx =
�
0
cross product
sinθ = 1: dl ⊥ r
Thursday, October 27, 2011
2π
�
�
µ0 Iadθ
a
µ0 Ia2
√
=
2 + a2 )3/2
2
2
4π x2 + a2
2(x
x +a
µ0 I
Bx component
Bx (x = 0) =
2a
Magnetic Field of a Wire Loop - on Axis
Thursday, October 27, 2011
iClicker: What is the Magnetic Field...?
direction at point P?
A. Into page.
Straight wires lead to no field at P.
Loop field is into the page.
B. Out of page.
C. Infinite magnitude and undefined direction, since it is along wire.
D. Right, the same direction as the current in the wire.
E. Up.
Thursday, October 27, 2011
Thank you.
See you Monday.
Thursday, October 27, 2011