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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.
y

B
I
x
According to our work in the previous lesson, the net force on a current loop is

F 

However, the loop does experience a torque,  !!
PHYS104 Review:

(1)


(2)
 
(3)
Torque is usually dependent on your choice of the rotation axis. The
"Newton II for Rotation"
exception is when the ____________________________________.
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.
y
I

B
L
SOLN:
H
Rotation
Axis
x
B.
Equation
We can rewrite our work on the rectangular current loop as
 
  I AB


A is the directional area. It is a vector whose magnitude is the area of the
current loop and whose direction 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?
y
I=2A
B = 6.00 T
SOLN:
x
C.

Magnetic Moment - 
For microscopic phenomena, it is usually impossible to independently measure I


and A . It is only possible to measure the product I A .




EXAMPLE 1: Determine the direction of the magnetic moment for the following current
loop.
I
EXAMPLE 2: Determine the direction of the magnetic moment for an electron traveling
counter clockwise in the circular orbit shown below:
v
EXAMPLE 3: The proton 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:

L
EXAMPLE 4: The electron is also known to have intrinsic spin. Determine the
direction of the magnetic moment for the electron shown below:

L
EXAMPLE 3: The neutron also has intrinsic spin. The neutron does have a specific
magnet moment as shown below. If the neutron is neutral, the how can it have a magnetic
moment?

L


D.
Potential Energy and Torque
Charge
Density
Radial
Distance
We show in a later section, that a magnetic field is produced when current flows
through 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).


N
Bl oop
B
S

N
I
S
In This Case We Have No Torque As Magnetic Fields Are Aligned

Bl oop

B
N
S

I
S
N
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 longer 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 and 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
U
You should see the similarity between our results in this section and our work on
the electric dipole earlier in the course.
EXAMPLE:
In Modern Physics, we learn that particles like the electron and proton are not free to
align their spin axis and consequently 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?

B


L

B

B

L


L
II.
Source of Magnetic Fields
The source of all magnetic fields is ________________ ___________________
(i.e. ____________________________).
III.
Ampere’s Law

ΔL

B
A.
Along any closed curev, the 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.
 Bparallel L   INet
0
Closed 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 want to find B

Step 3: Your fingers now point in the direction of B
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?
B
I
C
A
D
EXAMPLE 2: Find the magnetic field at points A and B for the wire shown below:
B
A
I
EXAMPLE 3: Use the Right Hand Rule to draw the magnetic field for the current
dipole below:
I
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 math required to explain
this phenomena is beyond this course)
B
μ0 I
2π r
C.
Infinite Selenoid (Coil)
Consider what happens to the magnetic field lines as we place two current loops
on top of each other.
Side View
Cross Section View
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 like 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.
.
IV.
Magnetic Force Between Two Conductors
When two conductors 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.
I1
d
F
F
I2
L
F  μ 0 I1 I 2
L 2πd