Download Lesson 16 - Magnetic Fields III

Lesson 16 - Magnetic Fields III
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,  !!
PHYS1224 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


Charge
Density
Radial
Distance
D.
Potential Energy and Torque
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.
Biot-Savart Law (Magnetic Equivalent of Coulomb's Law)
P
A.
r

ds
The magnetic field at point P due to an infinitesimal (point like) piece of wire can
be found by the equation:


dB  I d s 2r̂
4πr
0
Where 0 is the permeability of free space (constant of proportionality)
I is current
r is the distance from source to point where field is determined
r̂ is the radial unit vector.
B.
Physical Interpretation of Biot-Savart Law
We will now rewrite the Biot-Savart law slightly
 
dB   



 
I d s  r̂

0 4 π r 2 

and compare it the equation for the differential electric field due to a differential
point source:
  1 dq 
dE  
 r̂

2 
 ε 4πr
 0

1.
Both equations show the surface area of a sphere, 4 π r 2 . This is the geometric
signature of a point source.
2.
We see that o is a proportionality constant for magnetic fields in the same way
that (1/o) is the proportionality constant for electric fields.
3.


The term I ds sin θ  dq v sin θ is the source of the magnetism just like dq is
the source of the electric field.
Note: It is important that you understand the physical implications of an equation and
its origins. If you just try to remember the equations, you will find it difficult to
retain them for any length of time or to apply them. You are also defeating one of
the educational purposes of the course. You are supposed to be learning how to
analyze and solve problems.
C.
Total Magnetic Field Due To A Wire
For the whole wire, we have

 μ I 
r̂
B  dB  o  d sr
2
4π
wire
This is similar to the result that we had for the electric field is our previous work.


E   dE  1  dq
r̂
2
r
4
π

all charges
o
D.
Direction Of The Magnetic Field Produced By A Current
1.


From the Biot-Savart Law, we see that dB must be perpendicular to both ds and
r̂ .
Thus, the magnetic field produced by a wire of any shape will always be

perpendicular to a plane containing the wire and the point where B is to be
determined.
2.
Right Hand Rule For Finding Magnetic Fields (Sums Up Biot-Savart
Results)
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
BIOT-SAVART EXAMPLE:
Derive the expression for the magnetic field at point P due to the infinitely long straight
current carrying wire shown below:

ds
y
r

R
I
SOLN:
P
x
r̂