Download Lecture 6: Pre-reading Light, Photons, and MRI

Physics E-1bx: Pre-reading for Lecture 6
March 10, 2015
Lecture 6: Pre-reading
Light, Photons, and MRI
In this lecture we’ll begin to explore light as one of the most important electromagnetic
phenomena. Physicists had studied light for centuries without really understanding what it was:
Newton thought that light was made up of little particles; other scientists found that light exhibits
characteristics of waves. In 1865 James Clerk Maxwell showed that light is simply an
electromagnetic wave—a wave consisting of oscillating electric and magnetic fields, traveling
through space at the speed of light. The fundamental wave relationship, λf = v, between the
wavelength (λ), frequency (f), and wave speed (v) holds true for light, just as it does for all
waves. (For light, of course, this equation is often written λf = c, where c is the speed of light.)
We will explore many of the interesting wave properties of light a bit later on in the course.
The view that light is a wave was widely accepted even before Maxwell’s time, but in
1905 Einstein pointed out that the phenomenon of the photoelectric effect strongly suggests that
light is somehow made up of particles! These particles—later called photons—don’t act quite
like ordinary objects because they are fundamentally described using quantum mechanics. But
in many instances we can think of photons as little “packets” of light energy that travel in
straight lines, bounce off of objects, and bend (or refract) when they encounter an interface (like
the interface between air and glass). In other words, we can think of photons as traveling along
light rays: straight paths that can reflect or bend.
The reflection of light rays on a mirror, for instance, can be thought of as due to a force
that the mirror exerts on the photon, changing its momentum. Just as a ball will bounce off of a
wall due to the force of the wall, so will a photon bounce off of a mirror or other surface. If the
surface is flat (planar), then the surface can only exert a force perpendicular to the surface. As a
result, the parallel component of the photon’s momentum will be conserved in the interaction
with the mirror. As you’ll see in
class, this means that light rays
y
photon
bounce off of a mirror such that
the angle of incidence θi is equal
θi
θf
x
mirror
to the angle of reflection θr.
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A related phenomenon occurs when a photon travels from one transparent medium (such
as air) into a different transparent medium (such as glass). In this case, some photons will be
reflected, but other photons will travel
across the interface. The ones that
y
photon
cross the interface will “bend” at the
interface, as suggested by the diagram
x
at right. This is the phenomenon of
θ1
air
glass
refraction. It turns out that the wave
θ2
speed of light in glass is slower than the
speed in air or vacuum. This change in
wave speed turns out to be equivalent to a force—a change in momentum—perpendicular to the
interface. Just as in the case of the mirror, though, the parallel component of the momentum is
conserved. As we’ll see in class, we can use this fact to derive the relationship between the
change in wave speed and the “bending” or change in angle at the interface.
Next we’ll discuss how we can use magnetism to image inside the body using MRI.
Particles like the electron and the proton have magnetic dipole moments. They act like tiny “bar
magnets,” with a N pole and a S pole. We say that this arises from a property called spin—
because these particles also have angular momentum. However, as you’ll see in your
homework, it doesn’t make sense to think of an electron as a “spinning ball of charge”—because
it just can’t spin fast enough for that to make sense. So we should think of the magnetic dipole
moment and the angular momentum as simply intrinsic properties of these particles. The angular
momentum of protons and electrons always has a component equal to ± 12 ! , where ! is Planck’s
constant divided by 2π. Planck’s constant (h = 6.63 × 10–34 J·s) is the basic quantity of quantum
mechanics, so spin is fundamentally a quantum property of these particles.
According to quantum mechanics, a proton or electron in a magnetic field can have only
two stationary states, which are called “spin up” or “spin down.” These states are called
stationary because no properties of a particle in these states will change with time. As you might
guess, these states correspond to the cases when there is no torque on the magnetic dipole: either
the dipole is aligned with the field (spin up), or opposite to the field (spin down). In any other
state there will be a torque on the particle, and it cannot be stationary.
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These stationary states form the basic states of a particle in thermal equilibrium. The
more stable (spin up) state will predominate at equilibrium, but the difference is tiny: out of a
million protons, there will be only few excess spin up protons. Since there is an energy
difference ΔE between these states, a photon with the correct frequency (ΔE = hf) can “flip” the
spin from up to down. This frequency is known as the Larmor frequency, and it depends on the
strength of the magnetic field. Then the spin-down protons will relax back to equilibrium, and in
the process they will emit photons with the same Larmor frequency. Detecting these emitted
photons will turn out to be essential for MRI.
So let’s imagine that you (the patient) are inside an MRI machine. There is a huge
magnetic field pointing along the z axis (let’s say that runs from your toes up through your head).
Now you apply an electromagnetic wave (photons) at the Larmor frequency. Some of the spins
will flip, and they will return back to equilbrium through two different processes. These
processes are known as T1 and T2. It turns out that these rates of relaxation depend sensitively
on the local material: is it solid, liquid, or gelatinous? Does it have a high water content, or a
high fat content? In other words, the values of T1 and T2 can help to identify the type of tissue
located at a particular location inside the body.
So: the goal of MRI is to find out the value of T1 and the value of T2 at every point within
the patient. The problem is that if you just get a sample from the whole patient, then you get a
washed-out average over the entire body. The key innovation—which was awarded the Nobel
Prize—is the use of field gradients to selectively locate T1 and T2 at a single point.
We’ll explain these field gradients in more detail in class, but the basic idea is that
instead of applying a uniform magnetic field along the z axis, you could apply a field that is (for
instance) slightly stronger near the head, and slightly weaker near the feet. With this gradient
field, you can selectively flip the spins of protons only in a single “slice”—for instance, a slice
through the abdomen. The key is the quantum resonance condition: the photons must have a
frequency that precisely matches the Larmor frequency of the spins. By matching the
frequencies at a single position in the body, you can obtain information about the T1 and T2
relaxation times at a single point. And that can tell you if there’s a tumor, for instance, and
precisely where the tumor is located.
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Learning objectives: After this lecture, you will be able to…
1.
Identify the poles of a magnet, and sketch the magnetic field lines created by a permanent
magnet (like a bar magnet).
2.
Construct an analogy between magnetic poles and electric charges, and explain the
limitations of that analogy.
3.
Construct an analogy between magnetic field lines and electric field lines, and explain the
limitations of that analogy.
4.
Predict the most stable configuration of a magnetic dipole in uniform magnetic field, and
explain why two magnetic dipoles (bar magnets) attract or repel one another.
5.
Calculate and draw the magnetic field created by a current in a long straight wire, using
the right-hand rule to predict the correct direction of the field.
6.
Calculate the magnitude and direction of the magnetic force on a moving electric charge,
using the cross product and the right-hand rule.
7.
Calculate the magnitude and direction of the magnetic force on a long straight wire
carrying an electric current.
8.
Predict the direction of the magnetic force between two long straight wires, both with
currents flowing in them.
9.
Identify the magnitude and direction of the magnetic dipole moment created by a circular
loop of current.
10.
Calculate the energy of a magnetic dipole placed in an external magnetic field.
11.
Define the magnetic flux through a loop of wire, and calculate the flux for a given
magnetic field and loop (including the sign of the flux).
12.
Describe what you can do to change the magnetic flux through a loop, and decide
whether the magnitude of the flux gets larger or smaller for each type of change.
13.
Explain Faraday’s Law of Induction and how it relates changes in magnetic flux with
induced current in a loop.
14.
Use Lenz’s Law to determine the direction of the induced current in a loop.
15.
Calculate the magnitude and direction of the induced current in a loop of wire, given the
change in the magnetic flux and the resistance of the wire, using both Faraday’s Law and
Lenz’s Law.
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