orced Oscillations and Magnetic Resonance
Dec 26, 2015
Forced Oscillations and Magnetic Resonance
A Quick Lesson in Rotational Physics:
TORQUE is a measure of how much a force acting on an objectcauses that object to rotate.
Moment of Inertia, I, is the rotational analogue to mass.
Angular acceleration, , is the second derivative of angular position.
Newton’s Second Law of Motion:
Rotational Equivalent of Newton’s Second law:
xmmaF
I
Where: is the torque I is the moment of inertia is the angular acceleration
innercoilB
Diagram of the magnetic field due to both coils, along with the angle associated with the angular displacement from equilibrium.
HelmholtzB
There are three forces acting on the compass needle:
1) magnetic force due to Helmholtz coils
2) magnetic force due to inner coils
3) damping force due to friction between the compass needle and the holding pin
Since we are dealing with rotational motion, these forces are actually torques.
Constants and Variables
• B = magnetic induction due to Helmholtz coils
= damping constant• F = amplitude of the
driving field the driving
frequency
dipole moment of the compass needle
• I = rotational inertia of the compass needle
angular displacement from equilibrium
Newton’s Second Law of Motion(rotational)
net = driving field + restoring + damping
The torque due to the driving field is:
tFlddrivingfie cos
The torque due to the restoring field is:
Brestoring
The torque due to the damping force is:
damping
The Second Order Differential Equation
BtFI cos
net = driving field + restoring + damping
Which corresponds to Newton’s Second Law of Motion:
Dividing through by I, the rotational inertia, and rearranginggives:
tI
F
I
B
I cos
I
21 I
B 20
then we let...
Substituting and 1
2 into the differential equation yields:
tI
F cos20
21
We assume a particular solution:
tctcp sincos 21
Since we are dealing with a oscillatory function it makes senseto assume the most general oscillatory solution involving the two oscillatory functions, sin and cos.
Solving for c1 and c2 we get:
)()(
))((
41
22220
220
1
I
F
c
)()(
)(
41
2222
21
2
o
I
F
c
Letting:
41
22220 )( D
This is the denominator of the c1 and c2 solutions
Further calculations lead us to the following equation:
tD
I
F
tD
I
F
p
sin)(
cos))(( 2
122
0
We can rewrite the particular equation, byusing some simple trigonometry:
)cos( tRp
where
241
22220
41
22
2
2
2222
][
)(
I
F
I
F
Ro
Through further calculations we can rewrite our particular solution in terms of amplitude of the driving field:
)cos( tz
Fp
We want to do this, in order to use our experimental datadirectly in the equation. The ratio in front of the cosine functionis the amplitude of the compass needle.
222220
2 )()( Izwhere
Let’s introduce a new relationship,
2
z
FEosc
With any kind of wave motion the relationship of the oscillatoryenergy is directly proportional to the square of the amplitude of the motion.
F which is the amplitude of the driving field, remains constant, thus the relationship
2
z
FEosc
can lead us to the conclusion:
minmax zE
In other words, when z is at a minimum, the oscillatory energy of the compass needle is at a maximum
-When we take the derivative of z and set it equal to zero, that is when z is at a minimum.
-When z is at a minimum, E, oscillatory energy, is at a maximum.
-When E is at a maximum, the deflection of the compass needle is at a maximum, and we get resonance.
0)()(20 220
22
Id
dz
When solved for will yield:
2
22
02
2
)(
I
Using the fact that
f 2
We can make a substitution and come up with:
2
2222
24
II
Bf
Rearranging and making the substitution
R
NiB
125
8 0
Gives us the final equation
2
22
2
02
8**
125
2
Ii
IR
Nf
The following graph was generated from experimental data:
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-50
0
50
100
150
200
250
300
350A plot of frequency squared vs. current, from experimental data
I
f2
The graph is a line with the equation in the form of
BAif 2
Where f2 is the frequency measured, i is the current measured, A is the slope of the graph which is directly proportional to the ratio , and B is the y-intercept which is directly proportionalto
I/2
Thus, we now can determine from our experimental data that the magnetic dipole moment-rotational inertia ratio to be 3.66 * 10^6, and the damping constant 2.03 * 10^-5 .
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-8000
-6000
-4000
-2000
0
2000
4000
6000
8000A resonant solution to our second order differential equation
t
theta
(Current used 1A)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-5000
-4000
-3000
-2000
-1000
0
1000
2000
3000
4000
5000A non-resonant solution to our second order differential equation
t
theta
(Current used 1A)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-3000
-2000
-1000
0
1000
2000
3000Another resonant solution to our second order D.E.
t
theta
(Current used 5A)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-2000
-1500
-1000
-500
0
500
1000
1500
2000
Another non-resonant solution to our second order D.E.
t
theta
(Current used 5A)
The NMR for ethyl acetate, C4H8O2: