Lesson 16 - Magnetic Fields III

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18 Οκτ 2013 (πριν από 3 χρόνια και 9 μήνες)

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Magnetic Fields II


I.

Torque On A Current Loop In A Uniform B
-
Field


A.

Theory



Consider a rectangular current loop as shown below.













According to our work in the previous lesson, the net force on a current loop is









However,

the loop does experience a torque,
!!


PHYS1
04

Review:





(1)









(2)






"Newton II for Rotation"






(3)

Torque is usually dependent on your choice of the rotation
axis. The




exception is when the ____________________________________.


I

x

y


EXAMPLE:

Calculate the torque on the rectangular current loop above for a rotation axis that is
parallel to the x
-
axis and lies in the center of the loop.












SOLN:

I

x

y


L

H

Rotation

Axis

B.

Equation



We can rewrite our work on the rectangular current loop as









is the directional area. It is
a vector

whose
magnitude

is the
area of the


current loop

and whose dire
ction is the
direction pointed to by your right


thumb

when you
wrap your fingers

in the
direction of the current flow

in he

loop.




Although we derived the torque equation using a rectangular current loop, it is

valid for any shape of current loop.



EXAMPLE 1:

Redo the previous example using our new torque equation.

















EXAMPLE 2:

What is the torque on a 50
-
loop coil of radius 1.00
-
m in the problem below?









SOLN:


B = 6.00 T

x

y

I = 2 A

C.

Magnetic Moment
-




For micr
oscopic phenomena, it is usually impossible to independently measure I

and
. It is only possible to measure the product
.



















EXAMPLE 1:
Determine

the direction of the magnetic moment for the following current
loop.











EXAMPLE 2:
Determine the direction of the magnetic moment for an electron traveling
counter clockwise in the circular orbit shown below:












I

v

EXAMPLE 3:
The proto
n is known to have intrinsic spin. This means that it acts as if it
is a "little spinning top" as shown below even though nothing is spinning in the classical
sense. Determine the direction of the magnetic moment for the proton shown below:














EXAMPLE 4:
The electron is also known to have intrinsic spin. Determine the
direction of the magnetic moment for the electron shown below:














EXAMPLE 3:
The neutron also has intrinsic spin. The neutron does have a specific
magnet mo
ment as shown below. If the neutron is neutral, the how can it have a magnetic
moment?












D.

Potential Energy and Torque





Cha
rge

Density

Radial

Distance



We

show in a later section, that a magnetic field is produced when current flows

thro
ugh a wire. Thus, our current loop is a magnet (a dipole magnet to be precise)

and the torque is trying to align the magnetic field of the loop in the same

direction as the external magnetic field (just like two bar magnets).












I
n This Case We Have No Torque As Magnetic Fields Are Aligned











In This Case, The External Magnetic Field Will Apply A Torque



We would have to work on the current loop in order rotate the loop so that its

magnetic field was no long
er aligned with the external magnetic field. If we

release the current loop, the external magnetic field will do work on our current

loop to realign the fields. Thus, magnetic potential energy was stored in turning

the loop to the unaligned position an
d given up when the loop was realigned.




By choosing the zero potential energy reference point when the fields are

perpendicular, we have that the potential energy for a magnetic dipole in an

external magnetic field is










You should see the similarity between our results in this section and our work on

the electric dipole earlier in the course.


EXAMPLE:

I




N

S

N

S

I




N

S

N

S

In Modern Physics, we learn that particles like the electron and proton are not free to
align their spin axis and con
sequently their magnetic field to just any angle. This is
known as spatial quantization and can't be explained by classical physics. The figure
below shows the three possible states for an electron. Which has the highest energy?






































II.

Source of Magnetic Fields



The source of
all

magnetic fields

is ________________ ___________________



(i.e. ____________________________).


III.

Ampere’s

Law


A.

A
long any closed curev, t
he
sum of

the products of the infinitesimal lengths along
the path times the magnetic field component along the path is proportional to the
net currnt that penetrates the surface bounded by the curve.









Where


0

is the permeability of
free space (constant of proportionality)


I is current


While Ampere’s Law is always true, it is useful in finding the magnetic field only if the
system has a high degree of symmetry where
you

know the shape of
the
magnetic field
lines
so that
we can draw
your

curve along
a
constant
B
field line in order to remove B
outside the sum.



Right Hand Rule For Finding Magnetic Fields



Step 1:


Place thumb of right hand along the direction of the current



Step 2:

Wrap your fingers in to the point where you wa
nt to find



Step 3:
Your fingers now point in the direction of





EXAMPLE 1:
Find the magnetic field at points A, B, C, and D for the wire shown
below where the current is flowing out of the page. From
this example what can you say
about the shape of the magnetic field lines for a current carrying wire?
















EXAMPLE 2:
Find the magnetic field at points A and B for the wire shown below:
















I

A

B

C

D

I

A

B

EXAMPLE 3:

Use the Right Han
d Rule to draw the magnetic field for the current
dipole below:














B
.

Total Magnetic Field Due To A Wire


For
a current carying
,
we know experiemntally that the magnetic field pattern
forms concentric circles of constant magentic field. (The m
ath required to explain
this phenomena is beyond this course)






I

C.

Infinite Selenoid (Coil)


Consider what happens to the magnetic field lines as we place two current loops
on top of each other.





By placing many loops together, we can increase the strength of the magnetic
field inside the loops while reducing the strength of the magnetic field outside the
coils. This device is called a solenoid and is useful for storing energy in a
magnetic field l
ike a capacitor stores energy in an electric field.






We can apply Ampere’s Law to find magentic field inside a solenoid with N loops
and a length of L.







.






Cross Section View

Side View

IV.

Magnetic Force Between Two Conductors


When two cond
uctors carrying current are placed near each other, there will be a
mutual attraction between the two conductors if they carry current in the same
direction and a mutual repulsion if they carry current in the opposite direction.







I
1

I
2

d

L

F

F